m 


UNIVERSITY  OF  CALIFORNIA 
AT   LOS  ANGELES 


'WE 


A    TEXT-BOOK 


PLANE     STJKVEYING 


BY 


WILLIAM   G.    RAYMOND,   C.E. 

MEMBER   AMERICAN    SOCIETY    OF   CIVIL    ENGINEERS;    PROFESSOR  OK   GEODESY, 

ROAD    ENGINEERING,    AND   TOPOGRAPHICAL   DRAWING,    IN    THE 

REXSSELAER    POLYTECHNIC    INSTITUTE 


NEW   YORK  •:.  CINCINNATI  •:•  CHICAGO 

AMERICAN    BOOK    COMPANY 


COPYRIGHT,  1896,  BY 
AMERICAN  BOOK  COMPANY. 


RAYMOND'S  PL.  SURV. 
\v.  P.  i 


TA 

Kut 

PREFACE. 


THIS  book  has  been  prepared  to  meet  the  needs  of  those 
beginning  the  study  of  surveying.  The  subject  treated  is  a 
simple  one,  and  an  effort  has  been  made  to  make  its  presen- 
tation clear.  The  book  is  a  text-book,  not  a  treatise,  and  it 
is  hoped  that  the  teachers  who  use  it  will  find  it  possible  to 
devote  their  lecture  work  to  amplification,  rather  than  to 
explanation,  of  the  matter  it  embraces. 

So  far  as  seemed  necessary  the  plan  of  giving  first  the 
general  method  and  then  the  details  has  been  adopted,  at 
the  risk  of  some  repetition,  because  I  believe  this  to  be  the 
clearest  method  of  presentation.  A  special  effort  has  been 
made  to  render  clear  and  comprehensible  those  points  which 
an  experience  of  fourteen  years  of  practice  and  teaching  has 
indicated  to  be  the  ones  presenting  the  greatest  difficulties 
to  students.  Simpler  matters  have  been  left  to  the  student 
to  work  out  from  suggestions.  The  book  can  be  read  under- 
standingly  by  any  one  who  has  completed  Trigonometry, 
two  formulas  only  being  given  whose  derivation  requires 
anything  beyond.  These  may  be  derived  by  the  teacher  for 
such  students  as  are  sufficiently  advanced. 

Particular  attention  is  called  to  the  systematic  arrangement 
of  computations  in  Chapter  VI.  ;  to  the  article  on  the  slide 
rule  ;  to  the  discussion  of  practical  surveying  methods  in 
Book  II.  ;  to  the  full  treatment  of  coordinates  ;  to  the  large 
number  of  examples  ;  and  to  the  use  of  the  terms  "  latitude 
difference "  and  "  longitude  difference "  for  the  old  terms 
"latitude"  and  "departure." 


228321 


4  PREFACE. 

The  whole  general  scheme  of  terms  is  thought  to  be  much 
more  logical  than  that  heretofore  in  use  ;  and  in  this  I  have 
the  support  of  Professors  Merriman  and  Brooks,  who  have 
adopted  practically  the  same  nomenclature  in  their  "  Hand- 
book for  Surveyors,"  recently  issued. 

The  logarithmic  tables  are  from  Professor  C.  W.  Crockett's 
"Trigonometry,"  and  are  particularly  suitable  for  surveyors' 
use. 

I  am  indebted  to  many  persons  and  books  for  valuable 
assistance.  Especial  acknowledgment  is  due  to  Professor  H. 
I.  Randall  of  the  University  of  California,  who  drew  Plate  IV.; 
to  Mr.  J.  J.  Ormsbee,  Mining  Engineer,  who  drew  Plate  V.  ; 
to  Mr.  John  H.  Myers,  Jr.,  A.B.,  C.E.,  for  the  problems  on 
coordinates  and  for  many  suggestions;  and  to  Professors  R.  S. 
Woodward  of  Columbia,  and  Frank  O.  Marvin  of  the  Uni- 
versity of  Kansas,  for  valuable  suggestions.  Mr.  E.  R.  Gary, 
C.E.,  Instructor  in  Geodesy,  Rensselaer  Polytechnic  Institute, 
has  given  much  help  in  the  preparation  of  examples. 

I  also  acknowledge  my  indebtedness  to  the  following 
instrument  makers  for  the  use  of  cuts  :  Messrs.  Buff  &  Berger, 
Boston,  Mass.;  W.  &  L.  E.  Guiiey,  Troy,  N.  Y.;  Keuffel  & 
Esser  Company,  New  York ;  G.  N.  Saegmuller,  Washington, 
D.  C.;  L.  Beckman,  Toledo,  O.;  Mahn  &  Co.,  St.  Louis,  Mo.; 
F.  E.  Brandis,  Sons  &  Co.,  Brooklyn,  N.  Y.  The  principal 
instrument  cuts,  furnished  by  the  Messrs.  Gurley,  Keuffel  & 
Esser,  and  Mahn  &  Co.,  will  be  known  by  the  firm  name  on  the 
cut.  Those  of  Buff  &  Berger  are  Figs.  19,  20,  48,  107,  148, 
151,  and  153.  G.  N.  Saegmuller  furnished  Fig.  54.  All  of 
the  cuts  used  are  covered  by  copyright. 

The  book  is  submitted  to  my  fellow  teachers  and  students 
of  surveying  in  the  hope  that  it  may  prove  useful  to  them  in 
their  work. 

TROY,  N.  Y.,  Auerwr-  1896.  WILLIAM    G.    RAYMOND. 


CONTENTS. 


BOOK  I.     PRINCIPAL  INSTRUMENTS  AND  ELEMENTARY 

CHAPTER  OPERATIONS. 

INTRODUCTION     .        .        .        .        .        *.       -.        ...        ft 

I.     MEASUREMENT  OF  LEVEL  AND  HORIZONTAL  LINES       .  .       13 

Instruments  used    .         .        .         .         •     • ,  «.        .        .  ,-  .       13 

Methods          .        .      •  .        ...        .        ...  .18 

Errors  Involved      .        »        ,        .        .         .        .        .  .       22 

II.     VERNIER  AND  LEVEL  BUBBLE  .        .  .    ..        .        .        .        .      31 

Vernier 31 

Level  Bubble 35 

III.  MEASURING  DIFFERENCES  OF  ALTITUDE,  OR  LEVELING        .      40 

Instruments 40 

Use  of  the  Level     . 50 

Adjustments  of  the  Level .63 

Minor  Instruments 72 

Leveling  with  the  Barometer         ......       74 

IV.  DETERMINATION    OF     DIRECTION    AND    MEASUREMENT    OF 

ANGLES .77 

The  Compass 77 

Compass  Adjustments 79 

Use  of  the  Compass .83 

Magnetic  Declination .86 

The  Transit   .         .         .  "* 95 

Use  of  the  Transit 100 

Adjustment  of  the  Transit 108 

The  Solar  Transit  .         .         .         .         .         .         .         .         .116 

Adjustments  of  the  Solar  Transit 122 

Saegmuller  Solar  Attachment 123 

Meridian  and  Time  by  Transit  and  JSun        ....     125 

V.     STADIA  MEASUREMENTS     . 127 

VI.     LAND  SURVEY  COMPUTATIONS Ill 

Balancing  the  Survey 144 

Supplying  Omissions 1 19 

Areas 152 

Coordinates 156 

Dividing  Land        .         .         .         .         .         .         .         .         .163 

Model  Examples     ....                          ...  165 

The  Planimeter 172 

The  Slide  Rule                                                                              .  179 


6  CONTENTS. 

BOOK   II.      GENERAL  SURVEYING   METHODS. 

CHAPTER  PAGK 

VII.     LAND  SURVEYS 201 

Surveying  with  the  Chain  alone    .         .         .      f  .'   '    .  v      .  204 

Farm  Surveys         .        .        .        .        .        .  x     .        .        .  208 

United  States  Public  Land  Surveys 219 

City  Surveying       , 230 

VIII.    CURVES 238 

IX.    TOPOGRAPHICAL  SURVEYING 244 

Topography 244 

Simple  Triangulation 253 

.   Mapping 261 

The  Plane  Table 268 

X.     EARTHWORK  COMPUTATIONS 275 

Ordinary  Methods 275 

Estimating  Volumes  from  a  Map  ......  281 

XI.    HYDROGRAPHIC  SURVEYING 287 

Soundings 289 

The  Sextant 294 

Measuring  Velocity  and  Discharge         .         .         .         .         .  298 

Direction  of  Current 304 

XII.    MINE  SURVEYING 305 

Surface  Surveys 305 

Underground  Surveys 308 

Connecting  Surface  and  Underground  Work         .         .         .  316 

Mapping  the  Survey 320 

APPENDIX. 

I.    PROBLEMS  AND  EXAMPLES 322 

Chapter  I 322 

Chapter  III 324 

Chapter  IV .        .        .324 

Chapter  V       .        .        .        .        .        .'        .        .        .        .326 

Chapter  VI     .        .        .  • 326 

Coordinates 328 

Chapter  VIII 335 

Chapter  IX .         .         .336 

Chapter  X 338 

Chapter  XI     .        .        .        , 339 

Chapter  XII   .        ...      .        .  ' 340 

II.     THE  JUDICIAL  FUNCTIONS  OF  SURVEYORS       .        .        .        .341 

III.  THE  OWNERSHIP  OK   SURVEYS,  AND  WHAT  CONSTITUTES   A 

SURVEY  AND  MAP     . 351 

IV.  GEOGRAPHICAL  POSITIONS  OF  BASE  LINES  AND  MERIDIANS 

IN  PUBLIC  SURVEYS 357 

V.     TABLES ;       ....  361 

INDEX .        .  471 


BOOK   I. 

PRINCIPAL   INSTRUMENTS   AND   ELEMENTARY 
OPERATIONS. 


INTRODUCTION. 


1.    Preliminary  conceptions.     An  ellipse  of  axes  AB  and  CD 
(Fig.  1),  being  revolved  around  its  shorter  axis  C'Z>,  will  gener- 
ate the  surface  of  an  oblate 
spheroid  of  revolution. 

If  we  imagine  the  sea  to 
extend  underneath  the  sur- 
face of  the  earth  so  that  the 
visible  solid  portions  of  the 
earth  will  be,  as  it  were, 
floating  on  a  ball  of  water, 
the  shape  of  that  ball  will 
be,  approximately,  that  of  an 
oblate  spheroid  of  revolution. 
The  surface  of  this  ball  is  called  the  mean  surface  of  the  earth. 

The  shorter  axis  is  that  connecting  the  poles  ;  the  longer  is 
the  diameter  of  the  circle  called  the  equator.  In  the  case  of 
the  earth  these  two  axes  do  not  differ  much  in  length,  and 
hence  the  earth  is  usually  spoken  of  as  a  "sphere  slightly 
flattened  at  the  poles."  It  may  seem  strange  to  the  student 
that  a  difference  of  twenty-seven  miles  should  be  spoken  of  as 
a  slight  difference.  But  when  it  is  said  that  this  difference 
is  about  one  third  of  one  per  cent  of  the  length  of  the  longer 
axis,  the  meaning  is  clearer. 

The  lengths  of  the  two  axes  according  to  the  latest  deter- 
minations1 are: 


Shorter  or  polar  axis  .      . 
Longer  or  equatorial  axis 


41,709,790  feet. 
41,852,404  feet. 


1  Clarke's  spheroid  of  1880.  The  values  as  found  for  Clarke's  spheroid  of  1866 
are  those  generally  used  by  geodesists.  They  are:  shorter,  41,710,242  feet;  longer. 
41,852,124  feet. 


IQ  INTRODUCTION. 

If  a  plane  is  passed  through  an  oblate  spheroid  of  revolu- 
tion, parallel  to  its  shorter  axis,  it  will  cut  from  the  spheroid 
an  ellipse.  If  passed  parallel  to  the  longer  axis,  it  will  cut  a 
circle.  So  with  the  earth  :  a  plane  passed  parallel  to  the  polar 
axis  cuts  from  the  mean  surface  of  the  earth  an  ellipse,  while 
one  passed  parallel  to  the  equator  cuts  a  circle.  Hence  me- 
ridians of  longitude  are  ellipses,  and  parallels  of  latitude  are 
circles. 

The  surface  of  the  sea,  or  that  surface  extended  as  before 
mentioned,  forms  what  is  called  a  level  surface,  and  a  line 
lying  in  this  surface  is  a  level  line. 

A  line  perpendicular  to  this  surface  at  any  point  is  a  verti- 
cal line  for  that  point.  (A  plumb  line  at  any  point  is  a  vertical 
line  for  that  point.) 

A  line  perpendicular  to  a  vertical  line  is  a  horizontal  line. 

A  tangent  to  the  earth's  mean  surface  at  any  point  is  per- 
pendicular to  the  vertical  line  at  that  point,  and  hence  is  a  hori- 
zontal line  for  that  point. 

An  inclined  line  is  a  straight  line  that  is  neither  vertical 
nor  horizontal. 

A  vertical  plane  at  any  point  is  a  plane  including  the  verti- 
cal line  at  that  point. 

A  horizontal  plane  at  any  point  is  a  plane  perpendicular  to 
the  vertical  line  at  that  point. 

A  vertical  angle  is  an  angle  formed  by  lines  in  a  vertical 
plane. 

A  horizontal  angle  is  an  angle  formed  by  lines  in  a  horizon- 
tal plane. 

At  any  point  on  the  earth's  surface  there  can  be  but  one 
vertical  line,  but  there  may  be  an  indefinite  number  of  horizon- 
tal lines ;  there  can  be  but  one  horizontal  plane,  but  there  may 
be  an  indefinite  number  of  vertical  planes. 

If  water  collects  upon  the  earth's  surface  in  some  depression 
above  the  mean  surface,  as  in  a  lake  or  pond,  or  even  as  in  a 
small  glass,  and  if  the  water  is  still,  its  surface  will  be  nearly 
parallel  to  that  portion  of  the  mean  surface  of  the  earth  that 
is  vertically  below  it ;  hence  it  will  be  a  level  surface,  and  a 
line  drawn  on  it  will  be  a  level  line.  Such  a  line  will  be  longer 
than  the  corresponding  line  drawn  on  the  mean  surface  of  the 


SURVEYING  DEFINED.  H 

earth  between  the  verticals  through   the   extremities   of  the 
upper  line. 

The  visible  solid  parts  of  the  earth  above  the  mean  sur- 
face and  the  invisible  solid  parts  below,  make  up  a  very 
irregular  body.  It  is  customary  to  speak  of  the  visible  parts 
of  the  earth's  surface,  both  fluid  and  solid,  as  the  "  surface 
of  the  earth."  In  the  definition  in  Art.  2,  however,  this 
term  must  be  understood  to  mean  not  only  the  visible  parts 
of  the  earth's  crust,  but  also  those  parts  that  must  be  reached 
in  connection  with  the  operations  of  mining,  bridge  build- 
ing, or  other  engineering  works  that  extend  below  the  visi- 
ble surface. 

2.  Surveying  defined.  Surveying  is  the  art  of  finding  the 
contour,  dimensions,  position,  etc.,  of  any  part  of  the  earth's 
surface,  and  of  representing  on  paper  the  information  found. 

The  operations  involved  are  the  measurement  of  distances, 
—  level,  horizontal,  vertical,  and  inclined,  —  and  of  angles, — 
horizontal,  vertical,  and  inclined ;  and  the  necessary  drawing 
and  computing  to  represent  properly  on  paper  the  information 
obtained  by  the  field  work. 

The  drawn  representation  is  called  a  map.  It  may  be  a 
map  showing  by  conventional  signs  the  shape  of  that  part  of 
the  earth's  surface  that  has  been  measured  ;  or  it  may  be  simply 
an  outline  showing  the  linear  dimensions  of  the  bounding  lines, 
together  with  the  angles  that  they  make  with  the  meridian,  or 
with  each  other,  and  sometimes  the  position  within  the  tract  of 
structures,  roads,  or  streams. 

A  map  of  the  former  kind  is  called,  a  topographical  map, 
and  the  operations  necessary  to  its  production  constitute  a 
topographical  survey. 

A  map  of  the  latter  kind  is  a  land  map,  and  the  operations 
necessary  to  produce  it  constitute  a  land  survey. 

Either  one  of  these  surveys  is  a  geodetic  survey,  if  the  tract 
is  so  large  that  the  curvature  of  the  earth's  surface  must  be 
taken  into  account.  This  limit  is  supposed'  to  be  reached 
when  the  tract  is  greater  than  one  hundred  square  miles,  but 
many  surveys  of  tracts  of  much  greater  area  than  this  are  made 
without  considering  the  mean  surface  to  be  other  than  plane. 


12  INTRODUCTION. 

Such  surveys  are  of  course  inaccurate,  but  may  be  sufficiently 
correct  for  the  purpose  they  are  to  serve. 

A  plane  survey  is  one  made  on  the  assumption  that  the 
mean  surface  of  the  earth  is  a  plane,  above  which  is  the  irreg- 
ular visible  surface  broken  by  hills  and  valleys.  Almost  all 
land  surveys  are  plane  surveys.  Only  plane  surveys  will  be 
considered  in  this  book. 

In  plane  surveying  all  measurements  are  referred  to  a  plane. 

In  geodetic  surveying  all  measurements  are  referred  to  a 
sphere,  or  spheroid,  according  to  the  area  covered  and  the 
accuracy  desired. 

It  must  be  borne  in  mind  that  no  physical  measurements 
are  exact.  The  art  of  surveying  makes  it  possible  to  deter- 
mine that  a  field  of  land  contains  a  certain  area,  more  or  less, 
that  a  mountain  is  so  many  feet  high,  more  or  less,  that  a  mine 
is  so  many  feet  deep,  more  or  less,  etc.  That  is  to  say,  it  is 
physically  impossible  to  measure  exactly  either  distance  or 
angles.  The  precision  attainable  or  desirable  in  any  survey- 
ing operations  will  be  discussed  elsewhere  in  this  book. 


CHAPTER   I. 


MEASUREMENT  OF  LEVEL  AND  HORIZONTAL  LINES. 

3.  The  line  to  be  measured.  The  distance  between  two 
points  on  the  surface  of  the  earth  is  the  length  of  the  level 
line  joining  the  verticals  through  the  points.  If  one  of  these 
points  is  much  higher  than  the 
other  (further  from  the  mean 
surface),  there  may  arise  con- 
fusion as  to  which  of  several 
lines  is  meant  by  the  above 
definition.  In  geodetic  sur- 
veying it  is  customary  to  re- 
duce the  distance,  when  meas- 
ured, to  the  length  of  the  level 
line  lying  in  the  mean  surface, 
and  contained  between  the  ver- 
ticals through  the  points.  The  distance  as  measured  will  always 
be  approximately  the  length  of  the  level  line  lying  midway  as 
to  altitude  between  the  two  points ;  and  this  length  is  that 
used  in  plane  surveying.  The  length  of  this  line  is  obtained 
by  measuring  a  series  of  short  horizontal  lines ;  the  sum  of 
these  lines  approximates  to  the  length  of  the  required  level 
line,  just  as  the  regular  polygon  of  an  infinite  number  of  sides 
approximates  to  the  circle. 

Fig.  2  will  serve  to  make  the  above  statements  clearer. 


FIG.  2. 


INSTRUMENTS  USED. 

4.  Chains.  The  instruments  used  are  chains,  tapes,  and 
wooden  or  metallic  rods.  Chains  are  of  two  kinds  —  Gunter's 
chain  and  the  engineer's  chain.  These  chains  are  alike  in  form, 

18 


14       MEASUREMENT  OF  LEVEL  AND  HORIZONTAL  LINES. 


but  vary  in  the  length  of  the  links  and  the  length  of  the  entire 
chain.  In  Gunter's  chain  the  links  are  7.92  inches  long,  and 
in  the  engineer's  chain  they  are  12.00  inches,  or  one  foot,  long. 
With  this  exception,  one  description  will  apply  to  both. 

A  chain  consists  of  one  hundred  "  links  "  made  of  iron  or 
steel  wire.  Number  12  steel  wire  is  best.  Fig.  3  shows  the 
form  of  the  links.  A  link  includes 
one  of  the  long  pieces  and  two  or 
three  rings,  according  as  there  are 
two  or  three  rings  used  to  connect 
the  long  pieces.  The  rings  are  in- 
troduced to  enable  one  to  handle  the 
chain  more  readily.  Brass  tags  with 
the  proper  number  of  points  mark  the 
ten-link  divisions  from  each  end  to- 
ward the  center,  and  a  round  tag  marks 
the  center  or  fifty-link  division.  The 
handles  are  of  brass,  and  are  usually 
made  adjustable,  so  that  slight  changes 
in  the  length  of  the  chain  may  be  cor- 
rected. A  special  form  of  handle  is 
sometimes  used,  having  a  knife  edge 
on  which  is  filed  a  notch  indicating 
the  zero  of  the  chain  for  the,  day,  the  chain  being  compared 
daily  with  a  standard  kept  for  the  purpose. 

The  Gunter's  chain,  having  100  links  of  7.92  inches  each,  is 
66.00  feet  long,  and  the  engineer's  chain  is  100.00  feet  long. 
The  former  was  devised  by  Mr.  Edward  Gunter,  for  the 
purpose  of  facilitating  the  computations  of  areas  that  have 
been  measured.  Its  length  was  so  taken  that  10  square 
chains  make  one  acre.  It  is  the  chain  referred  to  in  the 
table  of  surveyor's  square  measure,  which  should  be  carefully 
memorized.  This  table  may  be  found  in  almost  any  arith- 
metic. 

In  all  surveys  of  the  public  lands  of  the  United  States  the 
Gunter's  chain  is  used,  and  all  descriptions  of  land,  found  in 
deeds  or  elsewhere,  in  which  the  word  "  chain "  is  used,  are 
based  on  this  chain.  It  is  not  convenient  for  use  in  con- 
nection with  engineering  works,  such  as  railroad  construction, 


FIG.  3. 


INSTRUMENTS   USED.  15 

canal  building,  bridge  building,  etc.,  where  the  unit  of  meas- 
ure is  the  foot,  and  hence  in  such  work  the  engineer's  chain 
is  used. 


5.  Tapes.     Steel  tapes  are  better  than  any  sort  of  chain  for 
most   engineering  work  and   for   all  fine  surveying.       These 
tapes  are  made  in  various  forms,  from  thin  ribbons  half  an 
inch  wide  to  flat  wires  about  one  eighth  of  an  inch  wide  and 
one  fiftieth  of  an  inch  thick.     The  ribbon  tapes  are  graduated 
on  the  front  to  feet,  tenths,  and  hundredths  of  a  foot,  or  to  feet, 
inches,  and  eighths,  and  on  the  back  to  links  of  7.92  inches. 
They  usually  come  in  box  reels  and  are  from  twenty-five  feet  to 
one  hundred  feet  long.     They  are  suitable  for  very  nice  work 
of  limited  extent,  and  particularly  for  measurements  for  struc- 
tures, such  as  bridges  and  buildings,  both  in  the  shop  and  in 
the  field.     They  are  not  suitable  for  ordinary  field  operations 
of  surveying,  because  they  are  easily  broken.     For  such  work 
the  narrower,  thicker  tapes  are  preferable.     These  may  be  ob- 
tained in  any  lengths  up  to  a  thousand  feet  or  more ;  but  the 
lengths  usually  kept  in  stock  are  fifty  feet  and  one  hundred  feet. 
They  are  graduated,  usually  to  ten  feet  and  sometimes  to  fifty* 
feet  only,  but  may  be  graduated  to  suit  the  purchaser.     For 
surveying  work   the   following   graduation  is   recommended : 
Graduate  to  feet,  numbering  every  tenth  foot  from  one  end 
of  tape  to  the  other,  and  not  from  each  end  to  the  middle, 
as  in  the  chain.     Have  the  tape  one  foot  longer  next  the  zero 
end  than  its  nominal  length  and  divide  the  extra  foot  into 
tenths.      If  it  is  required  to  make  measurements  closer  than 
to  tenths  of  a  foot,  carry  a  pocket  steel  tape  from  three  to  five 
feet  long  and  graduated  in  feet,  tenths,  and  hundredths.     Hun- 
dredths can  usually  be  estimated  with  sufficient  precision. 

6.  Reels.     For  the  narrow  tapes   there   are   a  number  of 
patterns  of  reels,  most  of  them  aiming  to  furnish  an  open  reel, 
of  form  convenient  to  go  in  the  pocket  when  not  in  use,  and  so 
constructed  as  to  enable  the  surveyor  readily  to  reel  the  tape. 
There  is  but  one  reel  that  has  come  to  the  author's  attention 
that  combines  all  three  of  these  requisites.      This  is  shown 


FIQ.  8. 


FIG.  U. 


INSTRUMENTS   USED.  17 

in  Fig.  6,  and  a  modified  form  in  Fig.  4.  Other  forms  of 
reels  are  shown  in  Figs.  5,  7,  8,  and  9.  Fig.  5  is  a  reel  for  a 
tape  from  300  feet  to  1000  feet  in  length.  Fig.  9  shows  a  tape 
fitted  with  a  spring  balance  for  measuring  the  pull  on  the  tape 
when  in  use,  a  level  to  show  when  the  tape  is  held  horizontal, 
and  a  thermometer  to  give  the  temperature.  The  necessity 
for  these  attachments  will  appear  hereafter.  Such  a  tape  is 
used  for  land  surveys  in  the  city  of  New  York. 

Tapes  should  always  be  kept  dry,  and  if  wet  by  use,  should 
be  wiped  dry  and  rubbed  with  a  cloth  or  leather  that  has  the 
smallest  possible  quantity  of  mineral  oil  on  it. 


7.  Linen  tapes.     In  addition  to  the  steel  tapes,  linen  and 
"  metallic "    tapes   are  used   for   rough    work.     The    ordinary 
linen   tape   is    well   known   to   everyone.     The  metallic  tape 
is  a  linen  tape  with  a  few  strands  of  fine  brass  wire  woven 
through  it.      The   linen   tape  is  subject  to  great   change  in 
length  with  changes  of  moisture  in  the  atmosphere,  is  soon 
stretched,  and  is  easily  worn  out.      The  metallic  tape  is  not 
so  subject  to  change  in  length  with  change  of  atmospheric  con- 
ditions; it  is  soon  stretched,  but  is  not  nearly  so  soon  worn  out 
as  is  the  linen  tape.     Both  these  tapes,  being  easily  stretched, 
soon  become  quite  inaccurate  for  any  but  the  commonest  kinds 
of  work,  where  the  measurements  are  short  and  need  not  be 
closer  than  to  the  nearest  tenth  of  a  foot.     They  are  gradu- 
ated in  feet,  tenths,  and  half-tenths,  and  on  the  reverse  side  in 
links  of  7.92  inches.     Sometimes  they  are  graduated  in  inches. 
They  are  sold  in  paper  or  leather  box  reels. 

8.  Rods.     While  some  rough  measurements  are  made  with 
the  ordinary  ten-foot  pole  or  a  similar  arrangement,  no  other 
surveying  measurements  are  now  made  with  wooden  or  metallic 
rods,  except  measurements  of  base  lines  in  connection  with  im- 
portant geodetic  surveys ;  in  these  the  rods,  usually  metallic, 
are  arranged  with  other  devices  into  a  very  elaborate  piece  of 
apparatus.      It  is  believed  that   the   narrow  steel   and  brass 
tapes  will  entirely  supersede  the  elaborate  base  apparatus  now 
in  use. 

H'M'D  SURV.  — 2 


18      MEASUREMENT   OF   LEVEL   AND  HORIZONTAL   LINES. 

9.  Pins.     These  are  used  with  the  chain  for  the  purpose  of 
marking  chain  lengths.     They  are  about  fourteen  inches  long, 
made  of  steel  or  iron  wire  somewhat  less  than  a  quarter  of  an  inch 
thick  (No.  4  to  No.  6  wire  gauge)  with  a  ring  at  one  end  into 
which  is  fastened  a  strip  of  cloth  to  insure  ready  finding  of 
the  pin  when  stuck  in  tall  grass  or  brush.      The  other  end 
is  pointed.     Eleven  of  these  pins  constitute  a  set.     They  are 
usually  carried  on  a  ring  like  a  large  key  ring,  or  loose  in 
the  hand. 

10.  Range  poles.     Poles  are  used  to  range  out  the  line  to 
be  measured.     They  are  usually  of  wood,  round  or  hexagonal, 
six  to  eight  feet  long,  tapering  from  the  bottom  to  the  top, 
shod  with  a  pointed  iron  shoe,  and  painted  red  and  white  in 
alternate  strips  one  foot  long.      Gas  pipe  is  sometimes  used, 
but  is  not  recommended,  because,  while  it  does  not  break,  and 
while  from   its  weight  such  a  pole   is  easily  balanced  on  its 
point,  it  is  also  very  easily  bent,  and  very  difficult  to  straighten, 
and  is  too  heavy  to  be  handled  with  ease.    A  good  pole  for  nice 
work  in  cities  or  on  railway  surveys  is  made  of  hexagonal  steel 
about  three  eighths  to  five  eighths  of  an  inch  thick,  painted 
like  the  wooden  poles,  and  pointed  at  one  end. 

METHODS. 

11.  Preliminary  statement.     The  accurate  measurement  of 
a  line  on  a  comparatively  level  piece  of  ground  is  a  task  diffi- 
cult for  a  beginner  and  not  simple  for  an  expert  chaimnan, 
however  easy  it  may  seem.     The  method  of  doing  this  work 
on  ordinary  farm  surveys,  where  the  smallest  unit  of  measure 
is  the  link  (7.92  inches),  and  where  an  error  of  one  in  three 
hundred   to  one   in  five   hundred   may  be   tolerated,  will   be 
described ;    and  the  errors  incident  to  this  method  with  the 
necessary  corrections,  as  well   as   the   more   precise   methods 
applied  to  city  work,  will  then  be  discussed. 

12.  Chaining.     It  will  be  noticed  when  the  chain  is  received 
from  the  maker  that  it  is  so  folded  together  as  to  be  compact 
in  the  center  of  the  bundle  and  somewhat  bulky  at  the  ends,  in 


METHODS.  19 

shape  not  unlike  an  hour  glass.  This  results  from  doing  up 
the  chain  as  follows  :  Take  the  two  links  at  the  center  of  the 
chain  in  the  left  hand,  with  the  fifty-link  tag  on  the  left.  Take 
the  right  hand  ends  of  the  pair  of  links  next  but  one  to  those 
in  the  left  hand,  in  the  right  hand,  and  lay  the  right  hand  pair 
and  the  intermediate  pair  in  the  left  hand  diagonally  across  the 
pair  already  there.  In  like  manner  proceed  to  the  ends  of  the 
chain,  being  careful  always  to  place  the  new  links  diagonally 
across  the  links  already  in  the  left  hand  and  always  diagonally 
the  same  way.  It  is  better,  however,  to  do  up  the  chain  from 
one  end  instead  of  from  the  middle.  The  method  is  the  same 
except  that  the  two  end  links  are  first  taken  in  the  left  hand, 
the  handle  end  to  the  right.  A  little  more  time  is  required, 
but  the  chain  is  more  readily  loosened  for  service. 

It  will  be  assumed  that  the  ground  on  which  the  line  is  to 
be  measured  is  comparatively  level,  and  that  the  ends  of  the 
line  are  visible,  one  from  the  other.  If  there  is  no  visible 
object  to  mark  the  further  end  of  the  line,  a  range  pole  is 
placed  there,  toward  which  the  measurement  is  to  be  made. 
If  the  rear  end  of  the  line  is  also  marked,  the  head  chainman 
will  be  able,  without  difficulty,  to  put  himself  in  approximate 
line,  thus  saving  time.  The  strap  with  which  the  chain  is 
fastened  is  removed,  and,  if  the  chain  has  been  done  up  from 
the  middle,  the  two  handles  are  taken  in  the  left  hand  of  the 
forward  chainman  and  the  chain  bundle  in  the  right  hand, 
allowing  a  few  links  next  the  handles  to  fall  off.  The  chain 
bundle  is  then  thrown  out  in  a  direction  opposite  to  that  in 
which  the  measurement  is  to  be  made,  the  chainman  retaining 
the  handles  in  his  left  hand.  The  chain  should  be  thrown 
with  sufficient  force  to  straighten  it  out.  The  forward  man, 
usually  called  the  "head  chainman,"  then  takes  the  forward 
end  of  the  chain  and  the  pins,  and  starts  toward  the  further 
end  of  the  line,  while  the  rear  chainman  allows  the  chain  to 
slip  through  his  hands  to  see  that  it  is  not  kinked  or  bent.  If 
he  finds  any  bends  he  straightens  them.  If  the  chain  has  been 
done  up  from  one  end,  it  should  be  laid  down  near  the  starting 
point  with  one  handle  uppermost,  the  latter  to  be  taken  by  the 
head  chainman,  who  moves  off  toward  the  further  end  of  the 
line.  The  rear  chainman  allows  the  chain  to  slip  through  his 


20     MEASUREMENT   OF   LEVEL   AND   HORIZONTAL   LINES. 


FIG.  10. 


hands  as  before.  The  chain  gets  enough  rough  service  that 
can  not  be  avoided,  without  subjecting  it  to  the  additional  un- 
necessary wear  arising  from  throwing  it  out  forcibly,  to  be 
kinked  or  caught  in  the  brush  or  other  obstruction. 

One  pin  is  left  with  the  rear  chainman.  As  the  head  chain- 
man  walks  out,  he  places  one  pin  in  the  hand  that  carries  the 
chain,  the  remaining  pins  being  in  the  other  hand.  When  the 

chain  is  almost  out,  the  rear 
chainman  calls  "Chain."  The 
head  chainman  then  stops, 
turns,  and  straightens  the 
chain  while  being  put  into 
approximate  line  by  the 
rear  chainman.  The  chain 
being  taut  and  approxi- 
mately "lined,"  the  head 
chainman  assumes  the  posi- 
tion shown  in  Fig.  10,  and 
the  rear  chainman,  by  mo- 
tions or  the  words  "  right " 
and  "  left,"  accurately  aligns  the  pin  held  by  the  head  chain- 
man and  cries  "Stick."  The  head  chainman  then  forces  the 
pin  into  the  ground,  taking  care  that  it  marks  exactly  the  end 
of  the  chain,  and  cries  "Stuck." 

The  rear  chainman  then,  and  not  till  then,  draws  his  pin, 
keeping  hold  of  the  chain,  and  follows  the  head  chainman, 
who  moves  on  toward  the  forward  end  of  the  line,  and  the 
whole  operation  is  repeated.  After  one  pin  has  been  placed, 
the  head  chainman,  on  being  stopped  by  the  call  of  the  rear 
chainman,  can  quickly  put  himself  in  approximate  line  by 
sighting  back  over  the  pin  last  set  to  the  flag  left  at  the 
starting  point.  The  work  thus  proceeds  till  the  further  end  of 
the  line  is  reached,  when  the  head  chainman  walks  right  on 
past  the  point  till  the  chain  is  all  drawn  out.  He  then  returns 
to  the  point  and  notes  the  fraction  of  a  chain  between  the  last 
pin  and  the  point.  This  added  to  the  number  of  chains  gives 
the  distance  required.  If  the  distance  is  more  than  ten  chains, 
the  head  chainman,  when  he  sticks  his  last  or  tenth  pin,  calls 
"  Stuck  out."  He  then  waits  by  the  pin  till  the  rear  chainman 


METHODS.  21 

comes  up  with  the  pins  he  has  collected,  which  should,  with 
the  pin  he  started  with,  be  ten.  He  counts  them,  as  does  the 
head  chainman,  as  a  check,  and  they  note  one  "tally."  At  any 
time  the  number  of  tallies  plus  the  number  of  pins  in  the  rear 
chainman's  hands  gives,  in  tens  of  chains  and  chains,  the 
distance  that  has  been  measured. 

13.  Hints.     The  following  hints  may  be  of  service  to  be- 
ginners : 

The  rear  chainman  should  not  use  the  pin  to  brace  himself. 

He  should  hold  the  outside  edge  of  the  handle  flush  with 
the  rear  side  of  the  pin,  without  moving  the  pin. 

He  should  not  stop  the  head  chainman  with  a  jerk. 

He  should  not  sit  down  on  the  ground  while  holding  the  pin. 

Motions  and  words  should  be  sharp  and  distinct. 

Motions  and  instructions  should  be  proportionate  to  the  dis- 
tance that  the  pin  is  to  be  moved  ;  for  example,  the  arms  should 
not  be  swung  wildly  when  the  pin  is  to  be  moved  an  inch. 

The  head  chainman  should  see  that  the  rear  chainman  is 
looking  when  he  tries  to  straighten  the  chain. 

The  chain  should  not  be  jerked  in  straightening  it ;  it 
should  be  straightened  by  an  undulatory  motion. 

In  straining  the  chain,  the  head  chainman  should  pull 
steadily. 

Attention  to  these  matters  will  greatly  facilitate  the  work. 

14.  Chaining  on  slopes.     In  chaining  up  or  down  hill,  one 
end  of   the   chain   is   raised   till  both  ends  are  as  nearly  as 
possible  in  a  horizontal  line. 

If  the  slope  is  so  steep  that  one  end  of  a  full  chain  cannot  be 
raised  enough  to  bring  both  ends  in  a  horizontal  line,  the  chain 
is  "  broken,"  that  is,  the  distance  is  measured  by  using  a  part 
of  the  chain  at  each  measurement.  To  do  this,  the  chain  should 
be  carried  out  as  if  a  full  chain  were  to  be  used,  the  head  chain- 
man returning  to  such  a  point  on  the  chain  (preferably  a  ten- 
link  point)  that  the  portion  of  chain  between  himself  and  the 
rear  chainman  may  be  properly  leveled.  A  measurement  is 
made  with  this  portion,  then  with  the  next  succeeding  portion, 
and  so  on  till  the  whole  chain  has  been  used.  Care  must  be 


22     MEASUREMENT   OF   LEVEL   AND   HORIZONTAL   LINES. 

taken  not  to  get  the  pin  numbering  confused.  The  rear  chain- 
man  should  have  but  one  pin  for  the  whole  chain. 

The  high  end  of  the  chain  is  transferred  to  the  ground  in 
one  of  several  ways,  according  to  the  precision  desired.  If  the 
work  is  to  be  done  with  care,  a  plumb  line  is  used.  If  an  error 
of  a  tenth  of  a  link  in  each  chain  is  not  important,  a  pin  may  be 
dropped  from  the  high  end,  and  stuck  in  the  ground  where  it  is 
seen  to  fall.  The  pin  should  be  dropped  ring  down.  A  small 
pebble  will  serve  the  purpose  for  rough  work.  In  careful  work 
the  plumb  bob  should  not  be  dropped  and  the  pin  placed  in  the 
hole  made ;  but  it  should  be  noticed  where  the  bob  will  drop, 
and  the  ground  should  be  made  smooth  with  the  foot,  and  the 
bob  swung  down  till  it  is  still  and  just  clearing  the  ground ; 
then  it  should  be  carefully  lowered  till  it  touches.  The  chain- 
man  should  then  lower  his  grasp  on  the  string,  hand  over  hand, 
keeping  the  bob  steadily  in  its  place,  and  place  a  pin  in  the 
ground  at  the  point  of  the  bob.  The  pin  should  be  put  in  the 
ground  in  an  inclined  position  across  the  line,  so  that  the  point 
where  it  enters  the  ground  is  that  covered  by  the  bob.  The 
position  should  then  be  checked.  In  place  of  a  pin  a  small 
wire  brad  may  be  used  and  left  in  the  ground. 

In  chaining  up  hill,  the  rear  chainman  must  hold  the  bob 
directly  over  the  pin  which  has  been  set  in  an  inclined  position, 
and  must  at  the  same  time  align  the  head  chainman  and  see 
that  he  sticks  at  a  moment  when  the  bob  is  directly  over  the 
point.  It  will  be  at  once  inferred  that  it  is  easier  to  measure 
down  hill  correctly  than  up  hill.  Therefore,  where  close  work 
is  required  on  inclined  ground,  the  measurements  should  always, 
if  possible,  be  made  down  hill. 

ERRORS  INVOLVED. 

15.  Classes.  The  errors  involved  in  the  method  of  chaining 
just  described,  whether  the  work  is  done  with  a  chain  or  a  tape, 
are  of  two  classes :  (a)  constant  or  cumulative  errors,  and  (5) 
accidental  or  compensating  errors. 

(a)  Cumulative  errors  are  such  as  occur  each  time  in  the 
same  direction.  They  are  not  necessarily  equal,  but  may  be  so. 
Thus  a  line  so  long  as  to  require  that  a  chain  one  inch  too  short 
shall  be  applied  to  it  ten  times,  will  be  recorded  ten  inches  too 


ERRORS   INVOLVED.  23 

long,  the  error  of  the  chain  being  added  each  time  the  chain  is 
applied.     In  this  case  the  errors  are  equal. 

(J)  Compensating  errors  are  such  as  tend  to  balance ;  that 
is,  they  are  as  likely  to  be  in  one  direction  as  in  another.  Thus 
the  error  that  may  be  made  in  setting  the  pin,  if  it  is  attempted 
to  set  it  just  right,  will  be  a  compensating  error,  for  it  will  be 
set  ahead  of  the  true  point  about  as  often  as  it  will  be  set  behind 
it.  Error  in  plumbing  is  of  the  same  character. 

16.  Causes.  Cumulative  errors  arise  from  five  causes : 
(a)  erroneous  length  of  chain,  (6)  errors  in  judgment  in  mak- 
ing the  chain  horizontal  in  chaining  up  or  down  hill,  (c)  erro- 
neous alignment  of  the  chain,  (cf)  failure  to  straighten  the 
chain  for  each  measurement,  (e)  sag  of  the  chain  when  not 
supported  throughout  its  length. 

Compensating  errors  arise  from  accidental  inaccuracies  in 
setting  the  pin,  and  from  irregularities  in  the  pull  exerted  on  the 
chain  or  tape.  They  are  remedied  by  care,  and,  in  fine  work, 
by  measuring  the  pull  on  the  tape  by  a  spring  balance. 

Erroneous  length  of  chain  may  arise  from  any  one  of  six 
causes. 

(1)  One  or  more  links  may  be  bent,  making  the  chain  too 
short.     The  remedy  is  to  see  that  the  links  are  straight  or  to 
use  a  tape. 

(2)  Mud  or  grass  may  get  in  the  links  and  rings  with  the 
same  effect.     The  remedy  is  obvious. 

(3)  A  bent  link  that  has  been  straightened  has  been  per- 
manently lengthened,  thus  making  the  chain  too  long.     The 
remedy  is   to  compare  the   chain  or  tape  frequently  with  a 
standard  tape  kept  for  this  purpose.     If  the  chain  is  found  to  be 
slightly  too  long,  it  may  be  adjusted  by  the  nuts  at  the  handle, 
or  if  such  a  handle  as  is  described  in  Art.  4  is  used,  the  stand- 
ard length  of  the  chain  for  the  day  may  be  marked  on  the  handle. 

(4)  The  links  and  rings  wear,  thus  making  the  chain  too 
long.      While  the  wear  is  slight,  it  may  be  adjusted  at  the 
handle.     When  it  becomes  excessive,  it  must  be  known  and 
allowed  for  as  hereafter  described. 

(5)  The  chain  may  be  lengthened  by  too  hard  pulling,  but 
this  does  not  often  occur.     The  remedy  is  the  same  as  in  (3). 


^4     MEASUREMENT  OF   LEVEL   AND   HORIZONTAL   LINES. 

(6)  The  chain  may  be  too  long  or  too  short  according  as 
the  temperature  is  higher  or  lower  than  that  for  which  the 
chain  is  standard.  The  remedy  is  to  know  the  temperature  at 
which  the  chain  is  standard  and  that  at  wrhich  the  work  is 
done  and  make  the  necessary  correction  to  the  recorded 
measurements. 

In  general  it  may  be  said  that  erroneous  length  of  chain 
may  be  corrected  by  adjusting  the  handles,  or  by  comparing 
the  tape  or  chain  with  a  standard  and  correcting  the  records 
taken  according  to  the  errors  found.  It  should  be  carefully 
noted  that,  in  measuring  the  distance  between  two  points,  a 
long  chain  gives  the  distance  too  short  and  a  short  chain 
gives  the  distance  too  long,  while  in  laying  out  a  line  of 
given  length  the  errors  are  just  reversed.  Failure  to  appre- 
ciate this  difference  often  causes  confusion  and  error,  and 
hence  the  student  should  thoroughly  fix  it  in  mind.  Since 
similar  figures  are  in  area  as  the  squares  of  their  homologous 
sides,  the  erroneous  area  of  a  field  determined  from  measure- 
ments with  an  erroneous  chain,  will  be  to  the  true  area  as  the 
square  of  the  nominal  length  of  the  chain  is  to  the  square  of  its 
true  length. 

17.  Temperature.  The  coefficient  of  expansion  of  steel  is 
about  0.0000065.  (Tapes  and  chains  being  alike  subject  to  this 
error,  this  discussion  will  do  for  both.)  A  tape  or  chain  will 
expand  or  contract  sixty-five  ten-millionths  of  its  length  for 
each  Fahrenheit  degree  change  of  temperature.  Thus  a  line 
about  ten  chains  long,  if  measured  in  the  summer  with  the 
chain  at  a  temperature  of,  say,  80°  F.,  the  chain  being  stand- 
ard at  a  temperature  of  62°  F.,  will  be  recorded  0.117  links 
too  short ;  while  the  same  line  measured  with  the  same  chain 
in  midwinter  with  the  chain  at  a  temperature  of  0°  F.,  will  be 
recorded  0.403  links  too  long,  making  a  total  difference  of  0.52 
links  between  the  two  measurements.  This  is  an  error  of  one 
in  two  thousand  for  the  extreme  difference  in  temperature  of 
80°  F. 

It  is  thus  seen  that  for  all  ordinary  work  the  tempera- 
ture correction  may  be  neglected ;  but  in  city  work  where  an 
inch  in  frontage  may  be  worth  several  thousand  dollars,  it  is 


ERRORS   INVOLVED.  25 

very  necessary  that  the  temperature  be  determined  and  the 
standard  temperature  of  the  tape  known.  The  tape  shown  in 
Fig.  9  is  adjustable  for  the  effect  of  temperature.  A  scale 
numbered  to  correspond  to  the  thermometer  readings  indicates 
the  proper  setting  of  the  adjusting  screw.  The  spring  balance 
insures  a  constant  pull. 

18.  Sag.  The  effect  of  sag  in  shortening  a  tape  that  is  un- 
supported except  at  the  ends  is  given  by  the  following  formula 
in  which  I  is  the  unsupported  length  of  the  tape,  w  the  weight 
of  a  unit  of  length,  and  P  the  pull  in  pounds. 

_  I 
" 


This  formula  the  student  will  have  to  accept  until  he  has 
studied  the  elements  of  Mechanics  and  Calculus.  It  assumes 
that  the  tape  is  supported  only  at  the  ends,  and  that  it  is  stand- 
ard for  no  pull  when  supported  its  entire  length.  If  the  tape 
is  standard  for  a  pull  of  P0  pounds,  substitute  in  the  formula 
for  P  the  difference  P  -  PQ, 

If  the  tape  is  applied  n  times  in  measuring  a  line  and  each 
time  is  supported  only  at  the  ends,  and  the  pull  is  always  the 
same,  the  correction  for  the  whole  line  is  n  times  the  above 
expression. 

The  formula  gives  the  difference  between  the  length  of  the 
curve  of  the  unsupported  tape  and  its  chord.  The  real  dis- 
tance measured  is  the  chord,  while  the  distance  read  is  the 
length  of  the  curve  or  of  the  whole  tape.  It  is  evident,  there- 
fore, that  the  distance  is  read  too  long,  and  hence  the  formula 
is  a  negative  correction. 

19.  Pull.  If  the  chain  were  of  constant  cross  section  as  is 
a  tape,  the  amount  that  the  chain  would  stretch  for  a  pull  of 
P  pounds  would  be  given  by  the  following  formula  in  which 
I  is  the  length  of  the  chain  in  inches,  S  is  the  area  of  its  cross 
section  in  square  inches,  and  E  is  the  modulus  of  elasticity  of 
the  metal  of  which  it  is  made  : 

PI 
y=SE' 

1  This  formula  has  been  developed  by  Prof.  J.  B.  Johnson. 


26      MEASUREMENT   OF   LEVEL   AND   HORIZONTAL   LINES. 

E  for  steel  is  variable,  but  may  be  taken  at  28,000,000. 
There  is  no  such  thing  as  a  perfectly  elastic  material.  If  there 
were,  the  amount  that  a  given  length  of  the  material  would  be 
stretched  by  varying  pulls  would  be  proportional  to  the  pulls, 
and  supposing  the  piece  to  be  of  unit  cross  section,  as  one  square 
inch,  the  pull  that  would  stretch  it  by  its  own  length  is  known 
as  E,  the  modulus  of  elasticity.  For  any  other  than  a  unit  cross 
section  the  stretch  for  a  given  pull  will  be  inversely  propor- 
tional to  the  cross-sectional  area.  Hence  the  formula.  The 
lengthening  effect  of  a  given  pull  on  a  tape  would  be  as  in 
the  formula.  In  the  case  of  a  chain,  the  effect  would  be  some- 
what greater,  owing  to  the  elongation  of  the  rings. 

20.  Elimination  of  sag  and  pull.  To  find  the  pull  that  will 
just  balance  the  effect  of  sag,  equate  the  values  of  x  and  y 
and  solve  for  P.  Since  the  units  are  inches  in  the  y  formula, 
they  must  be  inches  in  the  x  formula,  and  I  must  be  the 
length  of  the  tape  in  inches,  and  w  the  weight  of  an  inch  of 
the  tape.  The  solution  gives 


whence 


A  good  practical  way  to  determine  this  value  is  as  follows  : 
Mark  on  a  smooth  level  floor  a  standard  tape  or  chain  length, 
with  the  tape  supported  its  entire  length,  and  with  only  enough 
pull  to  straighten  it.  Raise  the  tape,  and  supporting  it  only  at 
the  ends,  measure  with  a  spring  balance  the  pull  necessary  to 
bring  the  ends  over  the  marks  on  the  floor.  It  will  be  best  to 
have  one  end  fastened  in  a  firm  hook  in  the  wall  for  the  test, 
and  afterward  to  have  both  ends  held  by  the  chainmen,  that 
they  may  see  just  the  difficulties  involved.  The  test  should  be 
made  for  the  whole  chain,  the  half  chain,  and  the  quarter  chain. 
The  only  way  in  which  this  work  can  be  done  with  extreme 
nicety  is  by  employing  mechanical  means  to  pull  the  chain 
steadily,  and  a  telescopic  line  of  sight  to  transfer  the  floor 
marks  upward  to  the  tape  ends.  As  in  all  work,  except  the 
measurement  of  base  lines  for  geodetic  surveys,  or  elaborate 


ERRORS   INVOLVED.  27 

triangulation  surveys  of  cities,  the  chain  or  tape  is  to  be  held 
in  the  hands  of  the  chainmen,  it  will  be  unnecessary,  except  for 
comparisons,  to  resort  to  the  nicer  methods.  Experiments  of 
the  kind  noted  above  will  demonstrate  that  the  whole  chain 
should  never  be  used  unsupported,  and  that  the  tape  is  by  far 
the  most  satisfactory  measuring  instrument.  In  rough  work, 
where  a  precision  of  one  in  five  hundred,  or  even  one  in  five 
thousand,  as  a  maximum  limit,  is  sufficient,  the  chain  may  be 
used.  In  close  work  requiring  a  precision  of  one  in  five 
thousand  and  upward,  the  tape  should  invariably  be  used. 

21.  Alignment.     Errors  due  to  inaccuracy  of  alignment  of 
the  chain  are  usually  not  great.     In  ordinary  work  no  great 
pains  need  be  taken  to  align  the  chain  within  an  inch  or  two, 
except  where  stakes  are  to  be  driven  on  -the  line.     In  close 
work,  of  course,  the  chain  should  be  correctly  aligned.1 

22.  Slope.     In  chaining  on  slopes,  errors  of  judgment  in 
making  the  chain  horizontal  are  eliminated  by  the  use  of  a 
level  tube   fastened  to  one  end  of  the  chain,  which  tube,  if 
properly  adjusted,  will  indicate  when  the  chain  is  horizontal. 
This  is  rarely  used  with  a  chain,  but  frequently  with  a  tape. 
Much  can  be  done  without  such  a  level  by  having  a  third  man 
stand  on  one  side  of  the  chain  and  compare  the  parallelism  of 
the  chain  and  the  horizon,  or  the   horizontal   lines  of   some 
building.     If  there  is  no  horizontal  line  visible,  he  can  still 
judge  better  from  the  side  as  to  the  horizon  tality  of  the  tape, 
than  can  the  chainmen  at  the  ends.     It  is  almost  always  true 
that  the  lower  end  of  the  chain  is  not  raised  high  enough, 
because  a  horizontal  line  on  a  hillside  extending  in  the  direction 
of  the  slope,  always  appears  to  dip  into  the  hill.     Hand  levels 
(see  Art.  52)  carried  by  the  chainmen  are  of  great  service  in 
hilly  country.     The    effect  of   neglecting  the  slope  entirely, 
which  is  also  the  correction  to  be  applied  if  the  line  has  been 
measured  on  the  slope  instead  of  in  horizontal  lines,  is  given 
in  Appendix,  Table  I.,  page  361. 

1  Let  the  student  compute  the  error  arising  in  a  ten-chain  line  from  placing  the 
end  of  the  chain  first  six  inches  on  one  side  of  the  line  and  then  six  inches  on  the 
other  side,  throughout  the  measurement. 


28     MEASUREMENT   OF   LEVEL   AND   HORIZONTAL   LINES- 

It  will  be  seen  that  the  error  caused  by  neglecting  a  slope 
of  five  in  one  hundred  is  about  one  in  one  thousand,  while  a 
slope  of  ten  in  one  hundred,  which  is  not  unusual  in  hilly 
country,  causes  an  error  of  one  in  two  hundred.  Fifty  feet 
in  one  hundred  is  about  the  steepest  slope  met  with  in  nature, 
aside  from  rock  cliffs,  and  the  error  here  is  more  than  one  in  ten. 

On  a  slope  where  close  work  is  required,  it  is  considered 
best  to  measure  along  the  slope,  keeping  the  tape  or  chain  sup- 
ported throughout  its  entire  length,  and  making  the  necessary 
reductions  when  the  line  has  been  measured.  The  reduction 
can  be  made  exactly  by  the  use  of  a  table  of  versed  sines  if  the 
angle  of  slope  is  known.  It  may  be  approximately  obtained 
from  Table  I.,  page  361,  by  interpolating  for  the  small  angles, 
or  it  may  also  be  approximately  obtained  by  the  use  of  the  fol- 
lowing formula  when  the  rise  in  a  tape  length  or  in  the  entire 
line,  if  it  is  of  uniform  slope,  is  known  : 

The  square  of  the  rise  divided  by  twice  the  known  side,  be  it 
base  or  hypotenuse,  gives  the  difference  between  the  base  and 
hypotenuse. 

Demonstration  :  Let  B  be  the  base,  H  the  hypotenuse,  and 
R  the  rise  ;  O  being  the  difference  between  B  and  H.  Then 
B  —  H  —  G  and  H  =  B  +  G.  Assuming  H  known,  there  is 
written 

H*-  (H-  (7)2=  #2. 

whence  °  =' 


Neglecting  <72  as  a  very  small  quantity,  there  results 

„      R* 
=  2H' 

Similarly  if  B  is  known,  there  may  be  written 


and  as  before,  neglecting  <72,  there  results 
Hence  the  rule  already  given 


ERRORS  INVOLVED.  29 

23.  Precision  to  be  obtained.  In  measuring  lines  the  degree 
of  precision  obtainable  should  be  known  by  the  surveyor.  The 
author  suggests  the  degrees  of  precision  mentioned  below  as 
those  that  should  be  attained  ordinarily  before  the  surveyor 
can  say  he  is  doing  good  work.  The  figures  given  do  not 
refer  to  the  absolute  lengths  of  the  lines,  involving  a  knowl- 
edge of  the  absolute  length  of  the  chain  or  tape,  but  merely  to 
the  probable  error  of  the  mean  of  two  measurements  of  the 
same  line  made  with  the  same  tape  under  different  conditions. 
Not  all  conditions  of  work  are  covered ;  but  only  such  as 
usually  exist.  The  surveyor  will  be  able  to  judge  as  to  how 
closely  the  conditions  under  which  he  is  working  at  any  time 
correspond  to  those  given. 

In  good,  fairly  level  ground,  good  work  will  be  represented 
by  differences  between  two  measurements  of  one  in  twenty-five 
hundred,  and  excellent  work  by  differences  of  one  in  five  thou- 
sand, assuming  the  work  to  be  done  with  a  chain.  These 
differences  give  the  probable  error  of  the  mean  value  as  one  in 
seventy-five  hundred  and  one  in  fifteen  thousand,  and  the  prob- 
able error  of  a  single  determination  rather  better  than  -£-$•$-$  and 
TT5W  On  hilly  ground,  rough  and  covered  with  brush,  one 
in  one  thousand  might  be  considered  good  and  one  in  five 
hundred  passable,  where  the  land  is  not  of  great  value.  These 
differences  give  the  probable  errors  of  mean  and  single  meas- 
urement as  goVo  to  Tinnf  and  2uW  to  ToW  respectively.  It 
should  be  remembered  that  the  value  of  the  land  measured,  or 
the  object  of  the  survey,  is  a  better  basis  for  judgment  as  to 
passable  work  than  the  conditions  under  which  the  work  is 
done.. 

In  work  in  large  cities  the  author  thinks  that  a  precision  of 
one  in  fifty  thousand  should  be  obtained.  That  is,  it  is  thought 
that  the  probable  error  of  the  mean  of  two  measurements  should 
not  be  greater  than  one  in  fifty  thousand.  This  will  require 
that  the  same  line  measured  under  totally  different  conditions 
as  to  weather  should  be  recorded,  after  the  necessary  correc- 
tions for  pull,  grade,  and  temperature  have  been  made,  both 
times  alike,  within  about  one  in  seventeen  thousand,  or,  in 
round  numbers,  three  tenths  of  a  foot  in  a  mile. 

When  but  two  observations  of  a  quantity  have  been  taken, 


30    MEASUREMENT   OF   LEVEL   AND   HORIZONTAL   LINES. 

the  probable  error  of  the  mean  is  ^  D,  where  D  is  the  differ- 
ence of  values  determined.  The  probable  error  of  either  of 
the  observations  is  0.47  D  or,  roughly,  \  D.  (See  any  treatise 
on  Least  Squares.)  This  supposes  that  all  cumulative  errors 
and  mistakes  have  been  eliminated  by  correction  and  that  only 
accidental  errors  remain. 

The  following  are  the  requirements  for  securing  a  precision 
of  one  in  five  thousand  and  one  in  fifty  thousand.  For  inter- 
mediate standards,  the  requirements  will  lie  between  those 
mentioned : 

For  a  precision  of  one  in  five  thousand,  using  a  tape,  no  cor- 
rections for  sag,  grade,  pull,  or  small  changes  of  temperature 
need  be  made.  The  tape  may  be  stretched  by  hand,  the  pull 
and  horizontality  being  estimated  by  the  tapemen.  The  plumb 
line  will  be  used  on  uneven  ground  as  in  close  chaining.  The 
temperature  of  the  air  may  be  compared  with  that  for  which 
the  tape  is  standard,  and  a  corresponding  correction  deduced. 

For  a  precision  of  one  in  fifty  thousand,  the  temperature  of 
the  tape  should  be  known  within  a  degree  or  two  Fahrenheit ; 
the  slope  should  be  determined  by  measuring  over  stakes  whose 
elevations  have  been  determined  by  a  level,  or  by  measuring 
on  ground  whose  slope  is  known.  The  pull  should  be  known 
to  the  nearest  pound,  and  hence  should  be  measured  with 
spring  balances.  If  the  tape  is  held  on  stakes,  the  sag  cor- 
rection must  be  considered.  The  work  may  be  done  in  any 
ordinary  weather,  but  is  best  done"  on  cloudy  days,  so  that  the 
temperature  of  the  tape  may  be  more  constant.  In  sunny 
weather  the  mercurial  thermometers  attached  to  the  tape  may 
indicate  a  very  different  temperature  from  that  of  the  tape.  If 
the  absolute  length  of  the  tape  is  not  known,  of  course  the 
absolute  length  of  the  line  is  not  determined. 


CHAPTER   II. 


VERNIER  AND  LEVEL  BUBBLE. 

24.  Before  proceeding  with  a  description  of  surveying  in- 
struments, it  is  necessary  to  describe  two  important  attach- 
ments common  to  almost  all  such  instruments.  These  are  the 
vernier  and  the  level  bubble. 


VERNIER. 

25.  Vernier.  This  is  a  device  for  reading 
degree  of  precision  than  is  possible  with  the 
finest  convenient  division  of  the  scale.  Thus  a 
scale  graduated  to  read  tenths  of  an  inch,  may 
be  read  to  hundredths  of  an  inch  by  the  aid  of 
a  vernier.  This  is  done  by  making  an  auxiliary 
scale  called  a  vernier,  with  divisions  one  one-hun- 
dredth of  an  inch  smaller  or  larger  than  those 
of  the  main  scale.  If  the  divisions  are  larger 
than  the  main  scale,  the  vernier  is  called  a  ret- 
rograde vernier  ;  and  if  the  divisions  are  smaller, 
it  is  called  a  direct  vernier.  The  reason  for  this 
distinction  will  appear  hereafter.  In  Fig.  11  !S 
is  a  scale  divided  into  inches  and  tenths.  F"is 
the  vernier  made  by  dividing  a  space  equal  to 
nine  of  the  small  divisions  of  the  main  scale 
into  ten  equal  parts,  thus  making  each  division 
on  the  vernier  one  one-hundredth  of  an  inch 
shorter  than  a  division  of  the  main  scale.  The 
first  division  line  of  the  vernier  falls  one  one- 
hundredth  of  an  inch  toward  zero  from  the 
first  division  line  of  the  main  scale.  If  then 
the  first  division  line  of  the  vernier  is  made  to 
coincide  with  the  first  line  of  the  main  scale 

31 


to  a  greater 


—  . 

0 
—  5 

- 

-v 

B 

— 

—10 

1  — 

^ 

\ 

—  o 

— 

o      .  

_ 

—  5 

2  — 

-  v 

*-i 

—  10 

FIG.  11. 


32  VERNIER  AND  LEVEL  BUBBLE. 

the  vernier  will  have  been  moved  one  one-hundredth  of 
an  inch.  Similarly  the  second  division  of  the  vernier  is  two 
one-himdredths  of  an  inch  toward  zero  from  the  second  line 
of  the  main  scale,  and  hence  if  the  vernier  is  moved  along 
till  the  second  line  of  the  vernier  coincides  with  the  second 
division  of  the  scale,  the  movement  has  been  two  one-hun- 
dred ths  of  an  inch,  and  so  on.  If  the  vernier  is  moved  till  the 
zero  is  opposite  some  other  division  of  the  scale  than  the  zero 
division,  the  first  line  of  the  vernier  will  be  one  one-hundredth 
short  of  the  line  of  the  main  scale  next  ahead  of  the  zero  of 
the  vernier ;  the  second  line  of  the  vernier  will  be  two  one- 
hundredths  short  of  the  second  line  of  the  main  scale,  and  so 
on.  If  the  vernier  is  moved  along  a  little  further  till,  say,  the 
fourth  line  of  the  vernier  has  been  brought  into  coincidence 
with  the  fourth  line  of  the  main  scale  ahead,  the  vernier  has 
been  moved  a  further  distance  of  four  one-hundredths  of  an 
inch.  Hence  to  tell  how  far  the  zero  of  the  vernier  has  moved 
from  the  zero  of  the  main  scale,  note  the  inches  and  tenths  on 
the  scale  from  zero  to  the  zero  of  the  vernier,  and  get  the  frac- 
tional tenth  expressed  in  hundredths  by  looking  along  the  ver- 
nier and  finding  the  division  that  coincides  with  a  division  of 
the  main  scale.  This  vernier  is  called  direct,  because  in  read- 
ing it  one  looks  forward  along  the  vernier  in  the  direction  in 
which  the  vernier  has  moved. 

Let  it  be  required  to  read  the  length  of  the  bar  B.  Place 
one  end  of  it  opposite  the  zero  of  the  main  scale  and  vernier. 
It  will  be  noticed  that  the  other  end  is  opposite  a  point  on 
the  main  scale  between  one  and  three  tenths  inches,  and  one 
and  four  tenths  inches.  Move  the  vernier  till  the  zero  is 
opposite  this  end  of  the  bar.  To  read  the  length  of  the  bar, 
read  on  the  main  scale  one  and  three  tenths  inches  and  look 
along  the  vernier  and  find  that  the  sixth  division  coincides 
with  a  division  of  the  scale  and  that  therefore  the  length  of  the 
bar  is  one  and  thirty-six  one-hundredths  inches.  It  will  be 
observed  that  the  divisions  of  the  vernier  are  one  tenth  of  one 
tenth  of  an  inch  smaller  than  the  divisions  of  the  scale.  That  is, 
the  value  of  the  smallest  division  on  the  main  scale  divided  by 
the  number  of  divisions  of  the  vernier  gives  the  smallest  reading 
that  may  be  had  with  the  vernier.  This  is  called  the  least  count. 


VERNIER. 


33 


Iii  the  retrograde  vernier  a  space  equal  to  a  given  number 
of  divisions  of  the  main  scale  is  divided  into  a  number  one  less 
on  the  vernier.  Thus  for  a  vernier  reading  to  hundredths  of 
an  inch  with  a  scale  graduated  to  tenths,  eleven  divisions  of 
the  scale  will  be  divided  on  the  vernier  into  ten 
spaces,  making  each  division  one  tenth  of  one 
tenth  of  an  inch  longer  than  those  of  the  scale. 
The  vernier  is  therefore  placed  as  shown  in  Fig. 
12,  back  of  the  zero  of  the  scale  instead  of  ahead 
of  it,  as  in  the  direct  vernier.  "Back"  and 
"  ahead  "  are  used  with  reference  to  the  direction 
in  which  measurements  are  to  be  made.  From 
the  portion  of  the  scale  extended  above  the  zero 
in  the  figure,  it  will  be  seen  that  the  first  line  of 
the  vernier  is  back  of  the  first  line  of  the  scale  by 
one  one-hundredth  of  an  inch,  the  second  line  by 
two  one-hundredths,  and  so  on.  The  principle 
of  operation  is  the  same  as  in  the  "direct  vernier, 
except  that  one  must  look  backward  along  the 
vernier  to  find  the  coinciding  line. 

A  vernier  to  read  angles  is  generally  used  when 
the  angles  are  to  be  read  to  the  nearest  minute  or 
less.  The  principle  of  construction  is  the  same 
as  for  linear  verniers.  A  vernier  to  read  minutes 
will  usually  occur  with  a  circle  graduated  to  read 
half  degrees.  If  a  space  equal  to  twenty -nine  of 
such  divisions  is  divided  on  a  vernier  into  thirty 
equal  parts,  each  division  of  the  vernier  will  be  one  thirtieth 
of  thirty  minutes,  or  one  minute,  less  than  a  division  of  the 
main  scale,  and  the  instrument  is  said  to  read  to  minutes.  If 
a  circle  is  to  be  read  to  the  nearest  twenty  seconds,  it  is 
usually  divided  into  twenty  minute  spaces,  and  a  vernier  must 
then  have  sixty  divisions,  since 


n       3  ' 

n  =  60  divisions. 


That  is,  fifty-nine  parts  of  the  scale  must  be  divided  on  the 
vernier  into  sixty  parts,  making  each  part  of  the  vernier  one 


R'M'D  SURV.  —  3 


34 


VERNIER  AND  LEVEL  BUBBLE. 


sixtieth  of  twenty  minutes,  or  one  third  of  a  minute,  less  than 
a  division  of  the  main  scale.  Figures  13,  14,  and  15  show 

three  double  verni-> 
ers.  They  are  called 
double,  because  there 
are  really  two  verni- 
ers in  each  figure,  one 
FlQ-  13'  on  each  side  of  the 

vernier  zero.  They  are  thus  arranged  so  that  angles  may  be 
read  in  either  direction,  the  circle  graduations  being  numbered 
both  ways  for  the  same  purpose.  The  student  should  deter- 
mine whether  the  first  two  are  direct  or  retrograde,  the  least 
count  of  each,  and  their  readings.  The  third  is  a  peculiar 


FIG.  14. 

pattern  found  ordinarily  only  on  compasses.  It  is  a  double 
vernier,  direct  as  to  division  (though  it  is  sometimes  made  ret- 
rograde), and  the  lower  left-hand  and  upper  right-hand  por- 
tions form  one  vernier.  It  is  used  where  there  is  lack  of  space 
to  make  the  ordinary  form.  To  read  an  angle  measured  to  the 
right,  read  on  the  scale 
to  the  last  division  be- 
fore reaching  the  zero 
of  the  vernier,  follow 
to  the  right  along  the 


*" 


FIG.  15. 


,  . 
vernier,     noting     the 

lower  line  of  figures  for  a  coinciding  line,  and  if  none  is  found, 
pass  to  the  extreme  left  end  of  the  vernier  and  look  along 
toward  the  right,  noting  the  upper  line  of  figures  till  a  coin- 
ciding line  is  found.  Thus  the  reading  of  the  vernier  in  the 
figure  is  355°  20',  or  4°  40'. 


LEVEL   BUBBLE.  35 


LEVEL   BUBBLE. 

26.  Description.  The  spirit  level  consists  of  a  glass  tube 
almost  filled  with  ether,  the  remaining  space  being  filled  with 
the  vapor  of  ether.  The  bubble  of  vapor  will  always  seek  the 
highest  point  in  the  tube.  If  the  tube  were  perfectly  cylindri- 
cal, the  bubble  would  occupy  the  entire  length  of  the  tube  when 
the  tube  is  horizontal,  and  the  same  thing  would  be  true  if  the 
tube  were  but  slightly  inclined  to  the  horizon,  thus  making  it 
impossible  to  tell  when  the  tube  is  in  a  truly  horizontal  posi- 
tion. The  tube  is,  therefore,  ground  on  the  inside  so  that  a 
longitudinal  section  will  show  a  circular  arc.  A  line  tangent 
to  this  circle  at  its  middle  point,  or  a  line  parallel  to  this  tan- 
gent, is  called  the  axis  of  the  bubble  tube.  This  axis  will  be 
horizontal  when  the  bubble  is  in  the  center  of  its  tube.  Should 
the  axis  be  slightly  inclined  to  the  horizon,  the  bubble  will  move 
toward  the  higher  end  of  the  tube,  and  if  the  tube  is  ground  to 
the  arc  of  a  circle,  the  movement  of  the  bubble  will  be  propor- 
tional to  the  angle  made  by  the  axis  with  the  horizon.  There- 
fore, if  the  tube  is  graduated  into  divisions,  being  a  portion  of 
the  circumference  of  a  very  large  circle  (so  large  in  fact  that 
the  arc  of  a  few  seconds  is  quite  an  appreciable  length),  it 
will  be  possible  to  determine,  within  the  limits  of  the  tube, 
the  angle  that  the  axis  may  make  at  any  time  with  the  hori- 
zon, provided  the  angular  value  of  one  of  the  divisions  of  the 
tube  is  known.  This  is  done  by  simply  noting  how  many 
divisions  the  center  of  the  bubble  has  moved  from  the  center 
of  the  tube. 

It  will  be  evident  that  divisions  of  uniform  length  will  cover 
arcs  of  less  angular  value  as  the  radius  of  the  tube  increases, 
and  also  that  the  bubble  with  a  given  bubble  space  will  become 
more  elongated  as  the  radius  is  increased.  Therefore  the  bub- 
ble is  said  to  be  sensitive  in  proportion  to  the  radius  of  curva- 
ture of  the  tube,  and  this  is  also  indicated  by  the  length  of  the 
bubble.  The  length  of  the  bubble,  however,  will  change  with 
change  of  temperature,  becoming  longer  in  cold  weather  and 
shorter  in  warm  weather.  In  the  best  class  of  tubes  there  is  a 
partition  near  one  end,  with  a  small  hole  in  it  at  the  bottom,  so 
that  the  amount  of  liquid  in  the  main  tube  may  be  regulated, 


36  VERNIER  AND   LEVEL   BUBBLE. 

thus  regulating  the  size  of  the  bubble.  This  is  necessary, 
because  independently  of  the  effect  of  long  radius  a  longer 
bubble  is  more  sensitive  than  a  shorter  one.  A  bubble  should 
settle  quickly,  but  should  also  move  quickly  and  easily. 

27.  Determining  the  angular  value  of  one  division.  There 
are  several  methods  of  determining  the  angular  value  of  one 
division  of  the  bubble  tube,  all  essentially  the  same  in  principle. 

The  axis  is  moved  through  a  small  angle,  and  the  move- 
ment of  the  bubble  is  recorded  in  divisions  ;  then  the  angular 
value  of  one  division  is  at  once  found  by  dividing  the  angle  by 
the  number  of  divisions  through  which  the  bubble  has  moved. 


FIG.  16. 

It  is  not  easy  to  measure  the  small  angle  exactly  but  it  is  not 
very  difficult  if  closely  approximate  results  are  sufficient. 

In  Fig.  16  is  shown  a  level  vial,  as  it  is  sometimes  called, 
resting  on  a  level  trier.  The  construction  of  the  level  trier  is 
perhaps  sufficiently  clear  from  the  cut.  It  consists  simply  of  a 
board  resting  on  a  knife  edge  at  one  end,  and  capable  of  being 
raised  or  lowered  at  the  other  by  means  of  a  screw  so  divided 
as  to  tell  the  angle  of  inclination  of  the  board.  The  screw  is 
called  a  micrometer  screw,  because  it  will  measure  a  very  small 
movement.  Suppose  the  pitch  of  the  micrometer  screw  is  one 
sixtieth  of  an  inch.  Then  the  divisions  on  the  vertical  scale 


LEVEL   BUBBLE.  37 

attached  to  the  movable  board  will  be  one  sixtieth  of  an  inch  ; 
so  that  a  single  revolution  of  the  screw  will  move  the  scale  past 
the  edge  of  the  screw  head  by  one  division.  If  the  circum- 
ference of  the  disk  head  of  the  screw  is  divided  into  one  hun- 
dred parts,  and  the  screw  is  turned  only  so  much  as  will  cause 
one  division  on  the  disk  to  pass  the  scale,  the  board  has  been 
moved  vertically  through  one  one-hundredth  of  one  sixtieth  of 
an  inch.  If  «now  the  length  of  the  bar  from  pivot  to  screw  is 
known,  the  angular  movement  of  the  bar  may  be  computed. 
Thus,  if  the  length  of  the  bar  is  eighteen  inches,  and  the  bar  is 
raised  so  that  one  division  of  the  scale  passes  the  micrometer 
head,  and  so  that  in  addition  ten  divisions  of  the  micrometer 
head  pass  the  scale,  the  linear  elevation  of  the  end  of  the  bar  is 

li  x  gV  inch  =  0.018+  inch. 

Since  there  are  206,265  seconds  in  an  arc  equal  in  length  to 
radius,  there  results  the  proportion,  in  which  x  is  the  angle 
in  seconds, 

x       =  0.018+ 
206265  '       18 

Whence  x  =  206.265  seconds. 

If  now  a  bubble  tube  were  resting  on  the  bar,  and  the  run  of 
the  bubble  were  observed,  for  the  above  movement,  to  be  ten 
divisions,  the  value  of  one  division  would  be  20.6  seconds. 

Example.  In  the  above  example  it  is  found  that  the  run  of  the  bubble  is 
one  inch.  Find  the  radius  of  curvature  of  the  bubble  tube. 

Other  methods  of  finding  the  angular  value  of  a  division  of 
the  tube  will  be  suggested  in  the  problems  on  Chapters  III. 
and  IV. 

Many  of  the  level  vials  found  on  compasses,  and  on  the 
lower  plates  of  many  other  instruments,  are  not  graduated,  are 
ground  to  short  radii,  and  not  uniformly,  and  hence  are  not 
fit  for  accurately  leveling  the  instruments  to  which  they  are 
attached  ;  but  such  bubbles  are  cheaper  than  others,  and  when 
placed  on  a  compass  or  other  instrument  not  intended  for  high- 
class  work,  they  are  sufficiently  precise  for  the  purpose  for 
which  they  are  used. 


38  VERNIER   AND   LEVEL   BUBBLE. 

28.  Principles.  The  proper  adjustment  of  a  bubble  on  an 
instrument  so  that  one  can  determine  when  the  instrument 
is  level,  depends  on  the  following  principles  : 

I.  If  a  frame  carrying  a  bubble  tube,  and  resting  on  two  sup- 
ports that  lie  in  a  level  line,  is  reversed  end  for  end  on  the  sup- 
ports, the  bubble  will  occupy  the  same  position  in  the  tube  for  both 
positions  of  the  frame. 

In  Fig.  17  it  is  seen  that  the  axis  of  the  tube  makes  the 
same  angle  with  the  horizon  in  both  positions,  and  the  same 
end  is  higher. 

Conversely,  if  a  frame  to  which  a  bubble  tube  is  rigidly 
attached  is  reversed  on  two  supports,  and  the  bubble  occupies 

the  same  position  in  its  tube 
for  both  positions  of  the 
frame,  the  supports  lie  in  a 
level  line,  or,  as  is  usually 
said,  are  level.  It  should 
be  noted  in  the  above  that 
the  bubble  is  not  necessarily 
Fl°-  17-  in  the  center  of  the  tube, 

but  merely  retains  the  same  position  in  the  tube  for  the  direct 
and  reversed  positions  of  the  frame. 

If  the  axis  of  the  tube  in  the  foregoing  cases  is  parallel  to 
the  line  joining  the  supports,  the  bubble  will  lie  in  the  center 
of  the  tube,  and  if  not  parallel,  the  deviation  of  the  bubble 
from  the  center  of  the  tube  will  be  that  due  to  the  angle 
between  the  line  of  support  and  the  axis  of  the  tube.  If  in  the 
latter  case  the  bubble  is  brought  to  the  center  of  the  tube,  the 
line  of  supports  will  make  an  angle  with  the  horizon  (be  out  of 
level)  equal  to  that  between  the  axis  of  the  tube  and  the  line 
of  supports.  If  now  the  frame  carrying  the  level  is  reversed, 
the  movement  of  the  bubble  will  be  twice  that  due  to  the  angle 
between  the  axis  and  the  line  of  supports.1  If  the  tube  is  now 
raised  at  one  end,  or  lowered  at  the  other,  till  the  bubble  has 
moved  halfway  back  to  its  former  position,  the  axis  of  the 
tube  is  made  parallel  to  the  line  of  support.  The  line  of 
support  may  now  be  made  level  by  raising  the  lower,  or  low- 

1  Let  the  student  make  a  diagram  showing  this. 


LEVEL   BUBBLE.  39 

ering  the  higher,  support  till  the  bubble  stands  in  the  center 
of  its  tube. 

If  two  levels  are  attached  to  a  plate  at  right  angles  to  each 
other  and  parallel  to  the  plate,  the  plate  will  be  level  when 
both  bubbles  are  centered.  If  the  tubes  are  not  parallel  to  the 
plate,  it  will  be  difficult  to  determine  when  the  plate  is  level,  as 
the  position  of  each  bubble  for  level  plate  must  be  determined 
by  trial.  If  the  tubes  are  so  fastened  to  the  plate  as  to  permit 
of  being  adjusted,  their  parallelism  may  be  tested  and,  if  neces- 
sary, corrected  by  the  method  of  this  article. 

II.  If  a  frame  carrying  a  level  tube  is  revolved  about  a  verti- 
cal axis,  the  bubble  will  maintain  a  constant  position  in  its  tube. 

For  the  axis  of  the  tube  maintains  a  constant  angle  with 
the  horizon. 

Conversely,  if  a  frame  carrying  a  bubble  tube  is  revolved 
about  an  axis,  and  the  bubble  remains  in  one  position  in  the 
tube,  the  axis  of  revolution  is  vertical.  If  the  constant  posi- 
tion occupied  by  the  bubble  is  the  center  of  its  tube,  the  tube  is 
horizontal,  and  consequently  perpendicular  to  the  axis  of  revo- 
lution. 


CHAPTER  III. 

MEASURING  DIFFERENCES  OF  ALTITUDE,  OR  LEVELING. 

29.  General  principle.     It  will  be  evident  that  if  by  any 
means  a  line  of  sight  may  be  made  to  revolve  about  a  vertical 
axis  to  which  it  is  perpendicular,  it  will  describe  a  horizontal 
plane.     Omitting  consideration  of  the  curvature  of  the  earth, 
a  rod  graduated  from  the  bottom  up,  and  held  at  any  point  on 
the  ground,  will  be  cut  by  this  horizontal  plane  at  a  distance 
above  the  ground  equal  to  the  height  of   the  line  of   sight 
above  the  ground  at  the  point  where  the  rod  is  held.     The  dis- 
tance above  the  bottom  of  the  rod,  as  indicated  by  the  gradua- 
tions on  the  rod,  is  called  the  reading  of  the  rod.     If  the  eleva- 
tion of  the  line  of  sight  above  some  assumed  base  or  datum,  as 
sea  level,  is  known,  the  rod  reading  subtracted  from  that  eleva- 
tion will  give  the  elevation  of  the  point  where  the  rod  is  held, 
referred  to  the  same  base.     Conversely,  if  the  elevation  of  the 
point  where  the  rod  is  held  is  known,  and  it  is  required  to  find 
the  elevation  of  the  line  of  sight,  it  is  done  by  adding  the  rod 
reading  to  the  elevation  of  the  point.      While  there  are  many 
details  to  be  considered,  such  as  the  curvature  of  the  earth,  the 
adjustment  of  instruments,  atmospheric  conditions,  etc.,  the 
above  contains  the  essential  principle  of  leveling. 

INSTRUMENTS. 

30.  General  description  of  level.     Any  instrument  used  for 
the  purpose  of  securing  a  horizontal  line  of  sight  may  be  called 
a  leveling  instrument;    or,  as  is  more  usual,  simply  a  level. 
There  are  three  comparatively  common  forms  of  levels,  shown 
in  Figs.  18-20. 

40 


Fie;.  111. 


INSTRUMENTS.  43 

Fig.  20  is  a  precise  level,  or  level  of  precision.  Fig.  18  is 
known  as  a  Y  level.  Fig.  19  is  a  dumpy  level.  The  most 
common  of  these  is  the  Y  level,  so  called  because  the  telescope 
rests  in  Y-shaped  supports.  The  instrument  consists  essentially 
of  a  telescopic  line  of  sight  and  an  attached  bubble  tube,  whose 
axis  may,  by  adjustment,  be  made  parallel  to  the  line  of  sight, 
so  that  when  the  bubble  is  in  the  center  of  its  tube  it  will  be 
known  that  the  line  of  sight  is  horizontal.  These  are  com- 
bined with  a  leveling  head  which  contains  the  vertical  axis,  and 
screws  on  to  a  tripod.  A  sectional  view  of  a  Y  level  is  shown 
in  Fig.  21. 

The  dumpy  level  is  so  called  because  of  its  short  telescope 
with  large  aperture. 

The  precise  level  is  simply  a  modification  of  the  Y  level, 
so  improved  as  to  make  it  capable  of  doing  work  to  a  greater 
degree  of  precision  than  can  be  obtained  by  the  use  of  either 
the  Y  or  the  dumpy  level.  The  dumpy  level  is  sufficiently 
precise  for  all  work  that  does  not  require  the  precise  level, 
and  it  is  considerably  cheaper  than  a  Y  level  of  the  same  make. 
From  the  standpoint  of  the  optician,  the  Y  level  is  the  more 
perfect  instrument,  because  of  its  many  easy  adjustments ;  but 
this  very  feature  is  to  some  extent  an  undesirable  one  .from  the 
standpoint  of  the  engineer,  who  wants,  for  all  ordinary  work, 
an  instrument  with  few  parts  to  get  out  of  adjustment.  The 
dumpy  level  can  not  be  so  easily  and  exactly  adjusted  for  col- 
limation  as  the  Y  level,  but,  as  has  been  before  stated,  it  is 
sufficiently  precise  for  all  work  not  requiring  a  precise  level. 
It  is  used  almost  altogether  by  English  engineers,  having  been 
invented  by  an  Englishman  named  Gravatt,  whence  the  level  is 
frequently  called  the  Gravatt  level. 

31.  Telescope.1  The  telescope  of  the  level  consists  of  a  bar- 
rel in  which  slide  two  tubes.  One  of  these  tubes  is  the  eye- 
piece tube  carrying  the  eyepiece  lenses  LLLL,  Fig.  21,  and 
the  other  is  the  objective  tube  carrying  the  objective,  or  object 
glass  0.  The  objective  tube  is  moved  in  and  out  by  means  of 
a  pinion,  which  works  in  a  rack  attached  to  the  sliding  tube. 
The  tube  is  made  to  move  in  the  axis  of  the  barrel  by  having 

1  For  a  discussion  of  the  principles  of  telescopes  see  any  good  book  on  Physics. 


44 


LEVELING. 


it  pass  through  the  ring  at  6Y,  which  is  accurately  centered  in 
the  barrel.  The  eyepiece  tube  moves  in  and  out  of  the  barrel 
at  the  other  end  in  a  similar  way,  and  is  centered  by  the  ring 
shown  at  AA.  Instead  of  a  rack  and  pinion  movement,  the 
eyepiece  should  be  moved  by  turning  it  around 
a  small  screw  extending  through  the  barrel  and 
into  a  helical  slot  in  the  eyepiece  tube.  This  is 
a  better  plan  than  the  rack  and  pinion  arrange- 
ment, because  the  eyepiece  is  but  seldom  changed, 
and  when  once  set  should  not  be  easily  disturbed. 
Some  instruments,  however,  have  the  rack  and 
~"Y-.  pinion  movement,  and  it 

i£]  is  preferred  by  some  sur- 

veyors. In  addition  to 
these  two  tubes  there  is 
at  R  a  ring  (shown  sep- 
arately in  perspective  in 
Fig.  22)  which  carries 
two  fine  wires,  one  verti- 
cal and  one  horizontal. 
These  wires  are  either 
spider  lines  or  fine  plati- 
num wires.  The  spider 
lines  are  more  common. 
This  ring  is  centered  in 
the  barrel  by  means  of  the  screws  at 
BB.  There  are  four  of  these  screws, 
called  capstan-headed  screws,  arranged 
as  seen  at  Win  Fig.  18. 
If  the  ring  is  to  be 
moved  to  the  right,  the 
screw  on  the  left  is  first 
loosened,  and  then  the 
screw  on  the  right  is 
tightened,  thus  drawing 
the  ring  over  to  the 
right.  Similarly  for  the  vertical  movement  of  the  ring  —  if 
the  ring  is  to  be  moved  up,  the  lower  screw  is  first  loosened 
and  the  upper  screw  is  then  tightened,  and  vice  versa.  The 


Fia.  21. 


FIG.  22. 


INSTRUMENTS.  45 

difference  between  this  telescope  and  the  ordinary  or  Galilean 
telescope,  is  in  the  introduction  of  these  wires  and  a  difference  in 
eyepiece  necessitated  by  them.  In  the  ordinary  telescope,  sucli 
as  is  found  in  field  glasses,  there  are  no  wires  and  it  is  impossi- 
ble to  say  to  what  particular  point  in  an  object  looked  at  the 
axis  of  the  telescope  is  directed.  Moreover,  it  is  useless  to 
put  such  wires  in  a  field  glass,  since  no  image  of  the  object 
looked  at  is  formed  in  the  tube  of  the  telescope.  With  an 
angle-measuring  instrument,  or  any  instrument  that  must  be 
pointed  to  a  definite  point,  it  is  indispensable  that  the  exact 
point  to  which  the  axis  of  the  telescope  is  directed  be  known. 
It  is  perhaps  inaccurate  to  say  that  the  direction  of  the  axis 
must  be  known,  for  any  other  fixed  line  in  the  telescope  would 
do  as  well,  provided  the  adjustments  hereafter  to  be  described 
could  be  made  with  that  fixed  line.  It  is  more  convenient  to 
have  the  fixed  line  at  least  very  close  to  the  axis  of  the  tele- 
scope, for  reasons  that  will  appear. 

The  imaginary  line  joining  the  optical  center  of  the  object 
glass  and  the  intersection  of  the  cross  wires  is  known  as  the 
line  of  collimation,  and  this  is  the  line  that  is  directed  to  the 
precise  spot  toward  which  it  is  desired  to  point  the  telescope ; 
or  rather,  in  the  level,  it  is  the  line  that  indicates  the  point 
towards  which  the  telescope  is  directed. 

The  area  seen  at  one  time  through  the  telescope,  or  rather 
the  angle  between  the  rays  of  light  from  the  extreme  edges 
of  this  area,  measured  at  the  instrument,  is  known  as  the 
field  of  view.  This  field  of  view  is  larger  as  the  magnifying 
power  of  the  telescope  is  smaller,  and  varies  from  about  one 
and  one  half  degrees  to  about  fifty  minutes.  The  former  is 
for  the  commoner  kinds  of  surveyor's  transits,  and  the  latter 
corresponds  to  a  magnifying  power  of  about  thirty-five  diam- 
eters, or  about  what  is  found  in  the  better  leveling  instruments. 

An  image  of  an  object  within  the  field  of  view  is  formed  at 
a  point  back  of  the  object  glass,  and  the  glass  is  moved  in  or 
out  till  this  image  falls  in  the  plane  of  the  wires.  This  is 
called  focusing  the  objective.  The  point  in  the  image  covered 
by  the  intersection  of  the  wires  is  that  toward  which  the 
telescope  is  directed. 

It  would  be  practically  impossible  to  tell  when  the  focusing 


46 


LEVELING. 


had  been  done,  were  it  not  for  the  eyepiece,  which  is  nothing 
more  nor  less  than  a  microscope  with  which  to  obtain  a  magni- 
fied view  of  the  wires  and  the  image  formed  by  the  objective. 
This  is  done  by  first  focusing  the  microscope  eyepiece  till  the 
wires  are  plainly  seen.  Then,  any  other  objects  in  the  same 
plane  with  the  wires  will  be  seen  at  the  same  time,  and  objects 
not  in  that  plane  can  not  be  distinctly  seen,  and  hence  it  may 
be  determined  when  the  image  formed  by  the  objective  is  in 
the  plane  of  the  wires,  and  also  what  point  in  the  field  of  view 
is  covered  by  the  intersection  of  the  wires. 

The  objective  will  need  to  be  focused  anew  for  each  differ- 
ent object  looked  at,  unless,  as  will  rarely  occur,  all  the  ob- 
jects viewed  are  at  the  same  distance.  The  eyepiece,  on  the 
other  hand,  since  it  has  to  be  focused  for  only  one  distance, 
need  be  changed  only  for  different  individuals ;  and  hence,  if 
only  one  person  is  to  use  the  instrument  for  a  long  time,  the 
eyepiece  may  be  focused  once,  and  not  again  disturbed  during 
the  time  it  is  used  by  this  person. 

Telescopes  should  be  corrected  for  spherical  aberration  and 
should  be  achromatic.1 

The  eyepiece  shown  in  Fig.  21  has  four  "lenses,  and  is  com- 
monly known  as  an  erecting  eyepiece.  The  image  formed  by 

the  objective  is  inverted,  and 
the  eyepiece  inverts  the  image 
so  that  the  object  appears  right 
side  up  or  erect.  This  eyepiece 
has  been  generally  used  in 
American  surveying  instru- 
ments because  of  a  supposed 
difficulty  in  the  use  of  one  that  shows  the  object  inverted. 
Such  an  eyepiece,  shown  in  Fig.  23,  has  two  lenses  less  than 
the  erecting  eyepiece,  and  consequently  absorbs  less  light  and 
secures  better  definition  of  the  object  viewed.  It  is  to  be 
preferred  for  all  surveying  instruments,  and  is  well-nigh  indis- 
pensable for  some  kinds  of  work ;  for  instance,  for  stadia 
measurements,  to  be  hereafter  described.  The  inconvenience 

1  Students  unfamiliar  with  these  terms  can  find  their  meaning  in  any  good  dic- 
tionary or  encyclopaedia,  in  any  text-book  on  Physics,  or  in  Baker's  "Engineer's 
Surveying  Instruments." 


FIG.  23. 


INSTRUMENTS. 


47 


attending  the  use  of  a  telescope  that  shows  objects  inverted  is 
largely  imaginary.  A  few  hours  with  an  inverting  glass  makes 
its  use  as  natural  as  the  use  of  an  erecting  glass.  Dumpy  levels 
and  precise  levels  are  almost  -always  inverting  instruments  ; 
while  the  Y  level,  the  most 
commonly  used  in  this  coun- 
try, is  almost  always  erect- 
ing. As  has  already  been 
stated,  there  now  seems  to  be 
little  reason  for  the  existence 
of  the  Y  level. 

32.  Leveling  Rods.  There 
are  three  common  patterns  of 
leveling  rods  and  an  innum- 
erable number  of  uncommon 
ones.  The  three  common 
patterns  are  shown  in  Figs. 
24,  25,  and  26.  Each  of 
these  is  made  in  two  pieces 
about  seven  or  less  feet  long. 
The  New  York  rod  (Fig.  24) 
and  the  Philadelphia  rod 
(Fig.  25)  differ  in  that  the 
Philadelphia  rod  is  so  gradu- 
ated as  to  be  easily  read  at 
ordinary  distances  by  the  lev- 
eler,  while  with  the  New  York 
rod  the  target  must  be  set  and 
the  reading  taken  and  called 
out  by  the  rodman.  The  tar- 
get of  the  New  York  rod  is 
provided  with  a  direct  vernier, 
usually  placed  below  the  cen- 
ter of  the  target.  This  causes 
some  confusion  to  the  begin- 
ner, who  has  been  taught  to 
read  the  scale  at  the  zero  of  Fm-  **•  FlG-  w-  FlG"  2G' 

the  vernier  and  the  fractional  reading  by  looking  along  the 


48 


LEVELING. 


vernier    in    the    direction    of    motion   for  the  coinciding  line. 
There  need  be  no  confusion  if  the  rodman  remembers  that  it 

is  the  center  of  (he  target 
that  is  set  by  the  leveler, 
and  not  the  zero  of  the  ver- 
nier. A  little  study  of  the 
vernier  will  show  him  that 
the  main  scale  is  read  oppo- 
site the  ten  of  the  vernier 
and  the  fractional  reading 
by  noting  the  number  of 

mjm    "•(                |     m  the  vernier  line  coinciding 

•^*                      M.\  with  a  division  of  the  scale. 

^^|      •  This  rod  is  graduated    to 

j^BH  ""^                  Wf  hundredths  of  a  foot,  and 

™  ' '  the  vernier   permits   read- 

8  ings  to  thousandths.  When 

M^\  il  reading  greater  than  6.5 

feet  is  required,  the  target 
is  clamped  at  6.5  and  the 
rod  extended.  There  will 
be  found  on  the  side  a  sec- 
ond vernier,  which,  when 
the  rod  is  closed,  reads  6.5 
feet  and  which  gives  the 
readings  when  the  rod  is 
extended.  This  vernier  is 
of  the  usual  construction. 
The  Philadelphia  rod 
target  has  no  vernier,  but 
a  tenth  of  a  foot  is  divided 
into  half-hundredths,  per- 
mitting a  direct  reading  of 
0.005  foot  and  by  estima- 
tion to  0.001  foot.  The 
target  is  set  at  seven  feet 
for  greater  readings  than 
seven  feet,  and,  if  the  rod  is 
to  be  read  by  the  leveler,  it  is  extended  to  its  full  length,  the 


A  •• 


FIG.  27. 


140 

INSTRUMENTS.  49 

graduation  then  being  continuous  from  bottom  to  top.  If  the  tar- 
get is  to  be  set,  the  rod  is  extended  until  the  target  is  in  the  line 
of  sight;  it  is  then  clamped,  and  the  rod  is  read  by  the  rodman 
by  means  of  a  scale  on  the  back. 

The  target  of  the  Boston  rod 
(Fig.  26)  is  fixed  to  the  rod.  Read- 
ings are  all  obtained  by  extending 
the  rod.  It  is  held  with  the  target 
end  down,  for  readings  less  than 
5.5  feet,  and  is  inverted  for  read- 
ings greater  than  this.  It  is  read 
altogether  by  vernier,  the  scales 
and  vernier  being  on  the  sides. 
It  is  read  to  0.001  foot.  It  is  the 
lightest,  neatest  rod  of  the  three, 
and  the  least  used.  The  Phila- 
delphia rod,  which  is  the  heaviest 
of  the  three,  is  the  most  used  be- 
cause of  the  fact  that  it  may  be 
quickly  read  by  the  leveler.  In 
the  great  majority  of  sights,  read- 
ings are  taken  to  0.1  foot  only;  and, 
on  turning  points  (hereafter  de- 
scribed), it  is  usual  to  read  to  0.01 
foot  only.  Readings  to  0.001  foot 
are  required  in  but  a  very  small 
percentage  of  work  done  with  the 
level,  even  on  turning  points  or 
bench  marks.  For  this  reason 
many  engineers  prefer  a  "  self-reading "  rod  without  target, 
and  made  in  one  piece.  Such  a  rod,  the  standard  of  the 
Lake  Shore  and  Michigan  Southern  Railroad,  is  shown  in 
Fig.  27. 

It  will  be  noticed  that  the  figures  are  so  made  as  to  mark 
the  divisions  into  hundredths. 

With  a  target  rod,  much  better  work  may  be  done  if  the 
target  is  painted  with  diamond-shaped  figures  instead  of  with 
quadrants,  as  is  customary.     The  target  may  be  set  more  pre- 
cisely if  the  wire  has  a  sharp  angle  to  bisect  or  sharp  point  to 
K'M'D   SURV. — 4 


FJQ.  28. 


.  T. 


50  LEVELING. 

cover,  than  if  it  is  to  be  made  coincident  with  the  edge  of 
a  dark  surface.  Professor  Baker  finds  that  at  300  feet  the 
error  of  setting  a  quadrant  target  may  be  about  0.002  of  a  foot 
or  more  while  that  of  setting  a  diamond-shaped  target  may  be 
a  little  over  0.001  of  a  foot.  Such  a  target  is  shown  in  Fig.  28. 

USE  OF  THE  LEVEL. 

33.  Adjustment.     The  level  has  been  said  to  be  an  instru- 
ment for  securing  a  horizontal  line  of  sight.     It  will  be  evident 
from  the  construction  of  the  ordinary  leveling  instruments  that 
this  may  be  accomplished  with  those  instruments  if  the  line  of 
collimation  and  bubble  axis  are  made  parallel ;  because,  if  this 
condition  exists,  and  the  bubble  is  brought  to  the  center  of  the 
tube,  the  line  of  sight  will  be  horizontal. 

This  introduces  the  idea  of  adjustment,  and  the  adjustment 
of  the  level  consists  essentially  in  making  the  line  of  collima- 
tion and  the  bubble  axis  parallel.  There  are  other  adjustments 
for  convenience,  but  this  is  the  only  necessary  one.  The  gen- 
eral discussion  of  the  adjustments  will  be  deferred  till  the 
use  of  the  adjusted  level  in  doing  simple  leveling  has  been 
explained. 

34.  Setting  up.     To  "set  up"  the  level  is  to  place  it  in 
position  for  leveling,  including  making  the  line  of  sight  hori- 
zontal.   The  level  is  an  instrument  that  is  rarely  set  "online," 
except  in  making  certain  adjustments.     It  is  placed  in  that 
position  that  will  command  the  greatest  possible  number  of 
points  whose  elevations   are   to   be  determined.     To  set  up, 
plant  the  legs  firmly  in  the  ground  with  the  leveling  plates 
approximately  horizontal.     Focus  the  eyepiece  on  the  wires. 
Bring  the  telescope  and  attached  level  over  one  set  of  diagon- 
ally opposite  leveling   screws  and,  by  the   screws,  bring  the 
bubble  to  the  center  of  its  tube.     Perform  the  same  operation 
over  the  other  set  of  screws.     This  will  to  some  extent  disturb 
the  former  work.     Therefore  turn  the  telescope  again  over  the 
first  set  of  screws  and  relevel ;  again  over  the  second  set,  etc.. 
till  the  instrument  is  level  over  both  sets.     If  the  instrument 
is  in  adjustment,  the  line  of  sight  will  now  be  horizontal  in 
whatever  direction  it  is  pointed. 


USE   OF   THE   LEVEL.  51 

35.  Differential  leveling.  To  determine  the  difference  in 
elevation  between  two  points,  both  of  which  are  visible  from 
a  possible  position  of  the  level,  set  up  the  level  in  a  position 
such  that  a  rod  held  on  either  point  will  be  visible.  Turn  the 
telescope  toward  one  point  and  read  a  rod  held  there  by  a  rod- 
man.  The  rodman  will  then  carry  the  rod  to  the  other  point, 
and  the  telescope  will  be  directed  toward  that  point  and  the 
rod  read.  The  difference  in  readings  will  evidently  be  the 
difference  of  elevation  required.  Care  must  be  taken  to  see 
that  the  bubble  is  in  the  center  of  its  tube  when  each  reading 
is  taken. 

If  the  elevation  above  some  base  surface  of  one  of  the 
points  is  known,  the  difference  of  elevation  applied  to  the 
known  elevation  gives  the  elevation  of  the  second  point.  This 
operation  is  capable  of  further  analysis,  thus  :  The  rod  reading 
on  the  point  of  known  elevation,  added  to  the  known  eleva- 
tion, will  give  the  elevation  of  the  line  of  sight,  and  is  there- 
fore called  a  plus  sight.  The  r.od  reading  on  the  second  point 
subtracted  from  the  elevation  of  the  line  of  sight  will  give 
the  elevation  of  the  second  point.  This  reading  is,  there- 
fore, called  a  minus  sight.  From  these  considerations  the  fol- 
lowing definitions  are  formulated : 

A  plus  sight,  or  reading,  is  any  reading  taken  on  a  point 
of  known  or  assumed  elevation  for  the  purpose  of  determining 
the  elevation  of  the  line  of  sight. 

A  minus  sight,  or  reading,  is  any  reading  taken  for  the 
purpose  of  determining  the  elevation  of  the  point  on  which 
the  rod  is  read. 

A  very  bad  custom  of  calling  plus  sights,  "backsights," 
and  minus  sights,  "foresights,"  has  prevailed  in  the  past. 
It  has  been  a  source  of  confusion  to  the  beginner  and  is 
illogical.  It  probably  arose  from  the  fact  that  the  work  in 
leveling  is  considered  to  proceed  from  the  point  of  known 
elevation  toward  the  point  of  unknown  elevation,  and  that, 
therefore,  plus  sights  are  taken  in  a  backward  direction,  and 
minus  sights  in  a  forward  direction.  This  is  not  always  true, 
as  will  appear  later,  and  hence  the  nomenclature,  "  backsight  " 
and  "  foresight "  is  ill-chosen.  It  is,  moreover,  true  that  when 
a  minus  sight  is  taken  in  what  may  be  considered  a  backward 


52  LEVELING. 

direction,  the  beginner  becomes  confused  and  applies  the  wrong 
sign.  Hence  the  terms  should  be  abandoned. 

If  consideration  of  curvature  is  neglected,  the  instrument 
should  be  set  midway  between  the  two  points,  in  order  to  do 
correct  work.  It  will  be  apparent  that  if  this  is  done,  the 
amount  of  the  curvature  of  the  earth,  for  half  the  distance 
between  the  two  points  will  be  added  in  the  plus  sight,  and 
subtracted  in  the  minus  sight ;  that  is,  each  reading  will  be 
too  great  by  the  curvature.  The  amount  of  this  curvature  is 
about  8  inches  in  one  mile  and  varies  with  the  square  of  the 
distance.  The  student  may  determine  the  effect  on  a  rod 
reading  when  the  rod  is  held  528  feet  from  the  instrument 
and  when  held  279  feet  away.  Three  hundred  feet  is  about 
as  great  a  distance  as  will  permit  a  definite  reading  of  the  rod 
with  the  average  level ;  though  in  work  requiring  no  great 
exactness,  much  longer  sights  may  be  taken. 

If  the  points  whose  difference  in  altitude  is  required  are  so 
located  that  rod  readings  can  not  be  had  on  both  from  one  posi- 
tion of  the  instrument,  an  intermediate  point  is  chosen  that 
may  be  used  with  the  first  one,  and  the  readings  are  taken 
on  the  first  and  on  the  intermediate  point.  The  difference 
in  altitude  between  the  first  and  intermediate  point  is  thus 
obtained.  The  level  is  then  moved  to  a  position  between  the 
intermediate  and  final  point,  and  their  difference  of  altitude  is 
determined.  The  two  differences  added  or  subtracted,  as  the 
case  may  be,  will  give  the  required  difference.  It  may  be  neces- 
sary to  introduce  several  intermediate  points.  The  work  is  sim- 
ply a  succession  of  operations  like  those  of  the  first  case. 

It  is  unnecessary  to  determine  the  differences  of  altitude  of 
each  set  of  intermediate  points,  as  the  difference  between  the 
sum  of  all  the  plus  sights  and  the  sum  of  all  the  minus  sights 
will  give  the  difference  of  altitude  of  the  first  and  final  point. 
The  student  should  show  this. 

The  intermediate  points  that  are  chosen  should,  if  the  work 
is  to  be  well  done,  be  firm,  definite  points,  as  the  projecting 
part  of  a  firm  rock,  the  top  of  a  peg  firmly  driven  in  the 
ground,  etc. 

When  extensive  differential  leveling  operations  are  to  be 
carried  on,  requiring  close  work  (as  the  careful  determination 


USE   OF   THE   LEVEL.  53 

of  the  altitude  of  an  observatory  or  other  point  involving  the 
carrying  of  levels  over  many  miles  and  the  introduction  of 
many  intermediate  "  turning  points "),  it  is  well  for  the  rod- 
man  to  carry  a  "  point "  with  him.  A  very  convenient  form 
is  shown  in  Fig.  29.  This  is  made  of  a  triangular  piece  of 
boiler  plate  about  three  sixteenths  of  an  inch  to  one  quarter 
of  an  inch  thick  and  about  five  or  six  inches  on  a  side.  The 
three  corners  are  bent  down  to  form,  as  it  were,  a  three- 
legged  stool,  and  a  round-headed 
rivet  is  set  in  the  center.  A  small 
hole  is  drilled  in  one  side,  in  which 
to  fasten  a  string  or  chain.  When 
used,  the  points  are  firmly  pressed 
into  the  ground  with  the  foot,  and 
the  rod  is  held  on  the  rivet.  In 

some  work,  notably  railroad  leveling,  this  kind  of  point  is 
not  suitable,  because  it  is  well  to  leave  every  turning  point 
so  that  it  can  be  again  found. 

In  leveling  down  or  up  a  steep  hill,  the  distance  from  the 
instrument  to  the  rod,  called  the  length  of  the  sight,  may  be 
greater  or  less  for  minus  sights  than  for  plus  sights.  This 
may  be  avoided  by  zig-zagging.  Distances  on  each  side  of 
the  instrument  are  made  nearly  enough  equal  by  pacing. 

The  form  of  notes  that  is  kept  in  differential  leveling  is  as 
simple  as  the  work.  There  are  three  vertical  columns,  one  for 
the  name  or  number  of  the  point  observed,  one  for  the  plus 
sights,  and  one  for  the  minus  sights.  The  readings,  both  plus 
and  minus,  taken  on  any  point,  should  appear  in  their  proper 
columns  opposite  the  number  of  that  point. 

36.  Profile  leveling.  A  profile  of  a  line  laid  out  on  the 
ground  is  the  bounding  line  of  a  vertical  section  that  includes 
the  line  whose  profile  is  desired.  It  shows  the  elevations  and 
depressions  along  the  line.  Profile  leveling  differs  from  differ- 
ential leveling  in  that  the  elevations  of  a  number  of  points 
along  the  line  whose  profile  is  required  are  obtained  from  a 
single  setting  of  the  instrument.  The  principle  is  the  same  as 
that  of  differential  leveling,  but  the  method  of  keeping  the 
notes  and  of  doing  the  work  is  a  little  different. 


54  LEVELING. 

The  line  whose  profile  is  required 


FIG.  30. 


is  first  marked  out  on  the 
ground  by  stakes  or 
other  marks  placed  at 
such  intervals  as  may 
be  necessary.  These  in- 
tervals are  usually  reg- 
ular, and  in  railroad 
surveys  are  generally 
one  hundred  feet.  In 
city  streets  the  interval 
may  be  fifty  feet.  In 
other  surveys  the  inter- 
val may  be  less  or  more, 
according  to  the  nature 
of  the  survey.  The 
object  is  usually  to  re- 
produce to  scale  on 
paper,  the  profile  de- 
sired. For  this  purpose 
profile  paper  is  gener- 
ally used,  on  which  the 
notes  are  plotted,  as 
will  be  described  later. 

Fig.  30  shows  a 
map  of  a  line  whose 
profile  is  desired,  which 
may  be  assumed  to  be 
the  center  line  of  a 
road.  It  also  shows  in 
exaggerated  form,  and 
to  no  scale,  the  position 
of  the  level  along  the 
road,  both  in  plan  and 
elevation. 

The  process  of  lev- 
eling is  as  follows  : 

Some  convenient 
point  is  chosen  as  a 
"  bench  mark,"  either 


USE   OF   THE   LEVEL. 


55 


because  its  elevation  above  some  accepted  datum  is  known,  or 
because  it  is  a  convenient  permanent  point  whose  elevation  may 
be  arbitrarily  assumed.  A  bench  mark  in  leveling  is  a  perma- 
nent point  of  known  or  assumed  elevation  from  which  leveling 
operations  may  proceed.  In  the  example  given,  the  B.  M.  is 
the  corner  of  the  water  table  of  a  building,  and  its  elevation  is 
assumed  to  be  1000.000  feet  above  some  arbitrary  datum  sur- 
face. The  elevation  of  the  B.  M.  should  be  so  assumed  as  to 
avoid  any  minus  elevations ;  that  is,  the  datum  surface  should 


LEVELS  ALONG  GRAFTON  ROAD,  QCACKENKILL 
TO  CROPSETVILLE. 

LEVELEK    TOUCEDA. 
ROD  HIGGINS. 

March  24,  1895. 

Sta. 

+3. 

II.  I. 

-S. 

Elev. 

B.M.No.l. 

0.462 

1000.4C2 

1000.000 

On  the  top  of  the  water  table  at  the 

0 

0.6 

999.8 

S.E.  cor.  of  the  brick  store  of  P.  Gosling. 

1 

2.2 

98.3 

Elev.  assumed. 

2 

4.3 

96.2 

+  50 

5.5 

95.0 

3 

5.2 

<t5.3 

4 

4.5 

96.0 

+  60 

3.8 

96.7 

5 

4.3 

96.2 

6 

6.0 

94.5 

7 

7.2 

93.3 

T.  P. 

3.602 

994.041 

10.023 

9::o.43<) 

On  stone  marked  +,  ten  feet  right  of 

8 

4.0 

990.0 

Sta.,  7  +  80. 

9 

5.3 

88.7 

10 

7.0 

87.0 

+  20 

7.2 

86.8 

11 

6.6 

87.4 

12 

6.2 

87.8 

13 

6.0 

88.0 

14 

5.8 

88.2 

15 

5.2 

88.8 

16 

5.7 

88.3 

FIG.  31. 

be  assumed  so  far  below  the  B.  M.  that  it  will  be  below  any 
point  reached  in  the  leveling. 

Having  selected  the  B.  M.,  write  in  the  notebook,  Fig.  31, 
under  the  column  marked  "Elev.,"  the  elevation  assumed,  and 
in  the  column  of  stations  mark  the  name  of  the  station  whose 


56  LEVELING. 

elevation  is  given,  in  this  case  B.  M.  No.  1.  On  the  right-hand 
page  write  a  description  of  the  B.  M.  so  clear  and  full  that 
another,  coming  to  the  place  long  after,  may  determine  with 
certainty  the  exact  spot  used. 

Set  up  the  level  at  a  point  convenient  to  the  B.  M.  and  to 
as  much  of  the  line  as  possible,  remembering  that  it  is  better 
to  make  sights  to  turning  points  approximately  equal.  Hav- 
ing the  level  set  up,  read  the  rod  held  on  the  B.  M.,  and 
record  the  reading,  in  the  column  marked  "  -f-S.,"  and  add  it 
to  the  elevation  of  the  B.  M.  to  get  the  height  of  the  line  of 
collimation,  which  record  in  the  column  headed  "  H.  I."  (height 
of  instrument).  The  rod  is  now  held  on  station  0,  and  is  read, 
and  the  reading  is  recorded  in  the  column  of  minus  sights,  op- 
posite station  0.  It  should  be  noted  that  it  is  usually  the  ele- 
vation of  the  ground  that  is  wanted,  and  not  that  of  the  top 
of  the  stake  that  marks  the  station.  Therefore,  the  rod  is  held 
on  the  ground,  and  not  on  the  stake.  The  rod  is  usually  read 
at  station  points  only  to  the  nearest  tenth,  as  it  would  be  folly 
to  try  to  determine  the  elevation  of  rough  ground  to  hundredths. 
Points  used  as  B.  M.'s  are  usually  read  to  hundredths  or  thou- 
sandths. Hundredths  are  generally  close  enough. 

The  rod  is  now  carried  to  station  1  and  read,  and  the  read- 
ing is  recorded  in  the  column  headed  "  — S.,"  opposite  sta- 
tion 1.  The  leveler,  if  he  is  quick,  may  make  the  subtraction 
of  the  minus  sight  on  one  station  while  the  rodman  is  passing 
to  the  next  station,  and  he  should  practice  doing  this.  He 
may  check  his  work  at  noon  or  night,  or  at  any  other  time 
when  the  survey  is,  for  any  reason,  temporarily  stopped.  The 
rod  is  now  read  on  station  2,  and  recorded  as  before.  Between 
station  2  and  station  3  there  is  a  decided  change  in  slope,  and 
the  rod  is  held  at  the  point  of  change,  the  rodman  either  meas- 
uring the  distance  from  station  2  to  the  point  or  estimating  it 
by  pacing,  according  to  the  importance  of  exactness  in  the  par- 
ticular survey.  He  calls  out  the  "plus"  as  (in  this  case)  "plus 
fifty,"  and  the  leveler  records  the  distance  as  shown  in  the 
notes,  and  then  reads  the  rod  and  records  the  reading  opposite 
the  recorded  distance  in  the  —  S.  column. 

It  will  be  noticed  that  a  number  of  minus  sights  have  been 
taken  with  the  instrument  pointing  backward  with  reference 


USE   OF   THE   LEVEL.  57 

to  the  direction  of  progress.  It  is  these  sights  that  are  some- 
times confused  with  "  backsights  "  when  the  terms  "  backsight " 
and  "  foresight  "  are  used. 

Thus  the  work  proceeds  till  the  rod  is  about  as  far  beyond 
the  instrument  as  it  was  in  the  beginning  back  of  it.  Not 
more  than  three  hundred  to  five  hundred  feet  on  either  side  of 
the  instrument  should  be  used,  if  fairly  good  work  is  to  be 
done.  A  point  is  then  selected  that  is  tolerably  permanent,  as  a 
solid  stone  or  a  peg  driven  into  the  ground.  This  point  need 
not  be  on  the  line  that  is  being  leveled,  and,  indeed,  should  not 
be  there,  if  the  line  is  the  center  of  a  traveled  road. 

The  rod  is  held  on  the  peg,  which  is  called  a  turning  point, 
or  simply  "peg,"  and  is  read  to  the  same  unit  as  on  the  B.  M., 
and  the  reading  is  recorded  in  the  column  of  —  S.,  opposite  the 
station  T.  P.  The  elevation  is  then  worked  out  at  once,  and 
the  level  is  carried  beyond  the  peg  a  convenient  distance  and 
again  set  up.  A  reading  is  then  taken  on  the  rod  held  on  the 
peg,  is  recorded  in  the  column  +S.,  and  is  added  to  the  eleva- 
tion of  the  peg  to  get  a  new  height  of  instrument,  which  is 
noted  in  the  proper  column  opposite  the  station  from  which  it 
was  determined.  A  description  of  the  peg  is  written  on  the 
right-hand  page,  so  that  it  may  be  again  found,  if  wanted  within 
a  reasonable  length  of  time.  The  work  then  proceeds  as  be- 
fore. After  an  interval  of,  say,  half  a  mile  or  a  mile,  a  perma- 
nent B.  M.  is  established.  It  may  be  used  as  a  turning  point, 
or  merely  determined  as  is  any  station,  except  that  the  reading 
will  be  taken  to  the  unit  used  for  B.  M.'s.  All  points  that 
are  likely  to  be  wanted  again  should  be  very  fully  described. 
Sketches  are  valuable  helps  to  words  in  making  descriptions. 

There  are  many  hints  that  could  be  given  to  facilitate  the 
work  of  leveling  ;  but  they  would  occupy  too  much  space,  and 
the  beginner  will  have  to  learn  details  from  experience.  One 
point  may  be  mentioned  :  Never  clamp  the  spindle.  It  is  un- 
necessary to  do  this  except  in  adjusting. 

Other  forms  of  notes  are  sometimes  used  by  engineers. 
Some  of  them  have  station  numbers  and  elevation  in  adjacent 
columns  ;  others  have  a  separate  column  for  minus  sights  on 
turning  points,  and  others  a  separate  column  for  elevations  of 
turning  points,  etc. 


58  LEVELING. 

37.  Making  the  profile.  The  profile  is  made  by  plotting 
to  scale  the  elevations  of  the  points,  at  the  proper  horizontal 
distance  apart,  and  connecting  the  plotted  points  by  a  line. 
Profile  paper  is  generally  used  for  this  purpose.  One  form  of 
ruling  for  such  paper,  known  as  Plate  A,  is  shown  in  Fig.  32. 
There  are  three  common  forms,  known  as  Plate  A,  Plate  B, 
and  Plate  C.  Plate  A  has  four  horizontal  divisions  and  twenty 
vertical  divisions  to  an  inch.  Plate  B  has  four  by  thirty  divi- 
sions per  inch  ;  and  Plate  C  five  by  twenty-five  divisions. 
Profile  paper  comes  in  rolls  twenty  inches  wide,  or  in  sheets. 

If  profile  paper  is  used,  a  convenient  elevation  is  assumed 
for  the  bottom  horizontal  line,  and  a  value  is  assigned  to  the 
space  between  successive  lines.  For  railroad  profiles  it  is 
usual  to  assume  one  foot  as  the  value  of  the  smallest  inter- 
val ;  the  paper  is  so  ruled  that  the  five-feet  and  twenty-five- 
feet,  or  fifty-feet  lines  are  accentuated.  A  smaller  value 
may  be  assumed  for  the  smallest  interval  when  it  is  necessary 
to  work  to  a  large  scale  and  to  show  minute  irregularities. 
It  is  usual,  also,  in  railroad  work  to  assume  the  value  of  the 
interval  between  successive  vertical  lines  to  be  one  hundred 
feet  or  "one  station."  Each  tenth  station  is  then  accentuated 
by  the  ruling.  This  scale  may  also  be  increased,  if  necessary. 
The  scales,  both  horizontal  and  vertical,  having  been  assumed, 
it  is  only  necessary  to  find  the  position,  horizontally  and  verti- 
cally, that  will  correctly  represent  any  point  whose  elevation 
and  distance  or  station  from  the  beginning  are  known,  and  to 
make  a  small  dot  at  that  place.  So  with  the  other  points. 
These  dots  may  then  be  connected  with  a  smooth  line  without 
angles.  It  may  be  necessary,  occasionally,  to  make  angles,  as 
at  the  bottom  of  a  steep-sided,  narrow  ravine  ;  but  ordinarily 
such  angles  do  not  occur  in  nature. 

The  following  method  of  making  the  profile,  assuming  the 
scale  as  small  as  described  for  railroad  work,  is  rapid.  The 
elevation  of  the  first  station  is  called,  and  a  point  is  made  on 
the  paper ;  the  elevation  of  the  second  station  is  called,  the 
pencil  is  put  down  on  the  paper  in  the  proper  place,  and  a  line 
is  drawn  back  to  the  first  point  without  making  a  decided  dot 
at  the  second  station.  The  line  is  drawn  freehand.  The  work 
then  proceeds  in  this  way.  It  seems  unnecessary  to  say  that 


USE   OF   THE   LEVEL. 


59 


turning  points  are  not  plotted,  unless  they  happen  to  be  crit- 
ical points  on  line  at  the  ground  level  —  a  rare  occurrence. 
The  notes  in  Fig.  31  are  shown  plotted  in  Fig.  32. 

If  profile  paper  is  not  used,  a  base  line  of  assumed  eleva- 
tion is  drawn  on  a  piece  of  paper,  the  horizontal  distances  to 
the  points  whose  elevations  have  been  determined  are  laid  out 
to  scale,  and  perpendiculars  are  erected  on  which  are  laid  off, 
to  vertical  scale,  the  elevations  of  the  points  above  that  as- 
sumed for  the  base  line.  The  points  thus  formed  are  connected 
by  a  line. 


A  convenient  scale  for  city  streets  is  : 

Horizontal,  1  inch  to  100  feet. 
Vertical,  1  inch  to  10  feet. 

Where  such  a  scale  is  used,  it  will  generally  be  better  to  con- 
nect the  points  found  by  ruled  lines. 

The  vertical   scale   is  usually  exaggerated,  except  in  pro- 
files made  for  detail  plans  of  buildings  or  other  works.     This 


60  LEVELING. 

is  necessary  to  make  the  irregularities  appeal  properly  to  the 
eye,  since  the  vertical  distances  are  so  small,  compared  with 
the  horizontal  ones,  that  when  a  long  horizontal  distance  is 
laid  out  to  small  scale,  what  would  be  considered  appreciable 
variations  in  altitude  on  the  ground,  will,  if  drawn  to  the  same 
scale  as  the  horizontal  distances,  appear  insignificant  on  the 
drawing. 

38.  Leveling  over  an  area.     Sometimes   it   is   required   to 
determine  the  elevations  of  a  number  of  points  in  a  compara- 
tively small  area,  as  when  a  city  block  is  to  be  graded  and  the 
quantity  of  earth  to  be  removed  or  supplied  is  required.     In 
this  case,  it  is  customary  to  divide  the  block  into  rectangles  or 
squares,  of,  say,  twenty  feet  on  a  side.     The  elevations  of  all 
the  corners  of  these  squares  or  rectangles  are  determined,  and 
after  the  grading   is   done,  the   same  squares  are  reproduced 
from  reference  stakes,  previously  set  outside  the  work,  and  the 
elevations  of   the  corners  are  again  determined.     The  differ- 
ence  of   elevation   at   any  corner,  before   and   after   grading, 
shows  the  depth  of  material  that  has  been  cut  away  or  filled 
in.     Thus  there  will  be  a  number  of   truncated  prisms,  the 
lengths  of  whose  edges  and  the  area  of  whose  right  sections 
are  known,  and  hence  the  volume  may  be  computed. 

In  such  a  case  as  this,  after  determining  the  elevation  of  the 
line  of  collimation,  readings  are  taken  to  as  many  corners  as 
may  be  seen  before  the  instrument  is  moved.  The  work  is  pre- 
cisely the  same  as  profile  leveling,  except  that  consecutive  read- 
ings are  not  necessarily  along  any  one  line. 

39.  Errors.     In  engineering  work   the  level  is  more  used 
than  any  other  surveying  instrument,  and  to  describe  the  de- 
tails of  its  use  in  the  various  surveys  in  which  it  finds  a  place 
would  be  beyond  our  purpose.     These  properly  belong  to  a 
description  of  the  methods  of  laying  out  and  executing  such 
engineering  works.     Some  hints  as  to  errors  arising  through 
carelessness  will  be  given.     A  discussion  of  minute  errors  in- 
volved is  necessary  only  in  a  treatment  of  precise  leveling. 

To  avoid  errors  of  adjustment,  the  effect  of  the  earth's  cur- 
vature and  refraction,  see  that  plus  and  minus  sights  are  of 


USE   OF   THE   LEVEL.  Ql 

equal  length.  See  that  the  turning  points  are  firm  so  that  there 
may  be  no  settlement  between  sights.  They  should  also  be 
such  that  the  rod  may  be  sure  to  rest  on  the  same  point  each 
time. 

The  bubble  should  be  in  the  middle  of  the  tube  when  the 
reading  is  taken  ;  therefore,  observe  just  before  and  after  the 


reading  to  see  that  the  bubble  has  not  moved.  On  unstable 
ground,  throwing  the  weight  of  the  body  from  one  foot  to  the 
other  may  throw  the  bubble  out.  Fine  work  can  not  be  done 
on  soft  ground.  A  mirror  similar  to  that  on  the  precise  level 
may  be  attached  to  a  dumpy  or  Y  level,  so  that  the  bubble  may 
be  observed  as  the  reading  is  taken.  Unequal  heating  of  dif- 
ferent parts  of  the  bubble  tube  will  cause  the  bubble  to  move 
toward  the  warmer  part.  This  may  be  shown  by  resting  the 
finger  lightly  on  the  tube,  to  one  side  of  the  bubble. 

Ordinary  work  must  be  done  in  all  kinds  of  weather,  and 
no  great  precaution  can  be  taken  to  avoid  the  effect  of  the  sun 
on  the  instrument. 

The  rod  should  be  held  vertical.  To  accomplish  this  on 
nice  work  a  "  rod  level "  is  used.  Fig.  33  shows  one  form  of 
such  a  level.  It  is  held  against  one  corner  of  the  rod.  AVav- 
ing  the  rod  back  and  forth,  from  and  toward  the  instrument, 
is  recommended  in  the  absence  of  the  rod  level.  When  this  is 
done,  the  lowest  reading  observed  is  used.  This  serves  very 
well  when  the  reading  is  several  feet ;  but  if  it  is  near  the  bot- 
tom of  the  rod,  and  the  rod  is  resting  on  the  middle  of  its  base, 
a  considerable  error  may  be  introduced.1  If  the  rod  is  rested 
only  on  its  front  edge,  the  method  is  correct. 

The  resultant  error  in  leveling  a  long  line  will  be  propor- 
tional to  some  function  of  the  distance  or  the  number  of  set- 

i  Let  the  student  show  why. 


62  LEVELING. 

tings  of  the  instrument,  or  both.  It  is  common  to  say  it  is 
proportional  to  the  square  root  of  the  distance.  With  readings 
taken  carefully  to  thousandths  of  a  foot,  excellent  work  with 
ordinary  levels  would  be  indicated  by  a  difference  of  0.01  foot 
to  twenty-four  settings  of  the  instrument ;  that  is,  if  a  line  of 
levels  is  run  for  twelve  settings  of  average  length,  and  rerun 
in  the  opposite  direction,  the  difference  in  elevation  of  the 
starting  point,  as  assumed  at  the  start  and  as  found  at  the  fin- 
ish, should  not  be  greater  than  0.01  foot.  If  it  is  as  much  as 
0.04  foot,  the  work,  if  errors  of  this  amount  are  important, 
should  be  rerun.  The  object  of  the  leveling  is  a  better  guide 
to  a  decision  as  to  what  is  good  work  than  any  arbitrary  limit. 
The  author's  students  ran  a  line  of  levels  from  San  Jose*, 
California,  to  Mount  Hamilton,  a  distance  of  twenty-seven 
miles,  and  were  required  to  repeat  the  work  if  the  error  of  any 
day's  run  proved  to  be  more  than  0.01  foot  for  twelve  settings 
of  the  instrument  in  advance.  The  difference  in  altitude  was 
about  4200  feet ;  and  at  one  or  two  places  along  the  line,  dif- 
ferences of  as  much  as  0.1  foot  were  observed,  while  the  dif- 
ference between  the  elevation  of  the  mountain,  as  determined 
by  the  advance  work  and  that  determined  by  the  back- 
ward work,  was  0.023  foot.  This  indicates  the  compensating 
character  of  the  errors  of  observation.  The  compensation  was 
probably  partly  due  to  the  fact  that  the  different  stretches  were 
run  by  different  men,  since  it  seemed  that  one  man  had  a  tend- 
ency to  make  errors  in  one  direction  and  another  to  make  them 
in  the  opposite  direction.  The  work  was  done  by  running  a 
half  day  ahead,  and  then  rerunning  the  same  stretch  backward. 

40.  Curvature  and  refraction.  The  effect  of  curvature  and 
refraction  is  shown  as  follows  :  In  the  exaggerated  figure  let  A 
be  a  point  in  the  line  of  sight  of  a 
level  telescope.  Let  FD  be  a  normal 
to  the  earth's  surface,  at  some  distance 
K,  —  approximately  equal  to  AB, 
GrF,  or  AD,  —  from  the  instrument. 
The  level  line  through  A  cuts  the 
normal  at  J5,  while  the  horizontal 
line  of  sight  cuts  it  at  D.  DB  is 


ADJUSTMENTS   OF   THE   LEVEL.  63 

then  the  correction  for  curvature  for  the  distance  K.  Radius 
OA  being  taken  equal  to  Gr  O,  we  have  by  geometry  and  a  small 
approximation  that  the  student  may  discover, 


The  effect  of  curvature  is  to  make  the  rod  read  high  and  hence 
to  make  the  point  F  seem  low. 

Refraction  has  the  opposite  effect,  that  is,  the  point  E  is  seen 
in  the  line  AD.  The  ray  of  light  from  any  distant  object  is 
bent  to  a  curve  that  is  irregular  because  of  the  irregularities  in 
the  atmosphere,  but  which  is  almost  always  concave  toward  the 
earth's  surface,  and  which  is  considered  for  average  conditions 
to  be  the  arc  of  a  circle  of  diameter  about  seven  times  that  of 
the  earth.  Therefore  the  effect  of  refraction  is  to  reduce  the 
correction  for  curvature  by  about  one  seventh. 

It  is  not  likely  that  this  value  is  even  closely  approximate 
so  near  the  ground  as  leveling  work  is  done.  The  combined 
effect  of  curvature  and  refraction  is  about  0.001  foot  in  220 
feet,  and  varies  as  the  square  of  the  distance. 

41.  Reciprocal  leveling.     So  long  as  plus  and  minus  sights 
are  at  equal  distances,  no  error  arises  in  the  work  because  of 
curvature  and  refraction  or  lack  of  adjustment  of  the  instru- 
ment.    It  not  infrequently  happens  that  sights  must  be  made 
of  unequal  lengths,  as  in  crossing  a  wide  river.     The  following 
method  avoids  the  errors  of  curvature,  refraction,  and  adjust- 
ment.    Establish  a  peg  near  each  end  of  the  long  sight.     From 
a  point  near  one  peg  read  a  rod  on  both  pegs.     Move  the 
instrument  near  the  other  peg  and  again  read  on  both.     The 
mean  of  the  two  differences  of  elevation  obtained  will  be  the 
correct  difference.1 

ADJUSTMENTS   OF  THE  LEVEL. 

42.  Focusing  the  eyepiece.     The  first  adjustment  (so  called) 
in  any  instrument  having  a  telescopic  line  of  sight  is  the  careful 
focusing  of  the  eyepiece  on  the  wires.     If  this  is  not  done, 
there  will  appear  to  be  a  movement  of  the  wire  over  the  image 

i  The  studeet  should  show  this  with  a  diagram. 


64  LEVELING. 

as  the  eye  is  moved  a  little  up  or  down  or  to  the  right  or  left. 
This  is  due  to  the  fact  that  the  wire  and  image  are  not  in  the 
same  plane.  The  eyepiece  is  best  set  by  turning  the  telescope 
toward  the  sky  and  moving  the  eyepiece  till  the  wires  appear 
sharp  and  black. 

43.  Adjustments  named.      In  addition  to  this  focusing   of 
the  eyepiece,  which  can  hardly  be  called  an  adjustment,  the 
adjustments  of  the  level  that  the  surveyor  is  usually  called 
upon  to  make  are  three  in  number : 

(1)  The  collimation  adjustment,  which  consists  in  making 
the  line  of  collimation  parallel  to  the  axis  of  the  telescope  tube. 

(2)  The  bubble  adjustment,  which  consists  in  making  the 
bubble  axis  parallel  to  the  line  of  collimation. 

(3)  The  Y  adjustment,  which  consists  in  making  the  axis 
of  the  Y  supports  that  carry  the  telescope  and  attached  bubble, 
at  right  angles  to  the  vertical  axis  of  the  instrument,  so  that 
the  level  when  leveled  in  one  position  may  be  turned  on  the 
spindle  180°  and  still  be  level. 

The  first  two  are  essential  to  correct  work.     The  third  is 
merely  for  convenience. 

44.  Collimation  adjustment.     To  make  the  first  adjustment, 
set  up  the  instrument  and  note  whether  the  vertical  wire  will 
coincide  with  some  vertical  line,  as  a  plumb  line  or  the  edge  of  a 
building.     If  not,  loosen  all  four  adjusting  screws  that  hold  the 
wire  ring,  and  turn  the  ring  till  the  condition  described  is  ful- 
filled.     Set  the  screws  and  unfasten  the  clips  that  hold  the 
telescope  in  the  Y's,  and,  by  means  of  the  leveling  screws  and 
the  horizontal  motion  of  the  instrument  about  its,  vertical  axis, 
bring  the  intersection  of  the  wires  to  cover  a  minute  distant 
point.     Clamp  the  horizontal  motion  and  make  the  coincidence 
of  the  wires  and  point  more  perfect  by  the  slow-motion  screw. 
Carefully  turn  the  telescope  over  in  the  Y's,  not  end  for  end, 
and  note  whether  the  intersection  of  the  wires  still  covers  the 
point.     If   not,  move  the  ring  that  carries  the  wires,  by  the 
capstan-headed  screws,  until  the  error  seems  to  be  one  half 
corrected.     This  is  done  by  correcting  first  one  wire,  then  the 
other.     Reset  the  wires  on  the  point,  fey  the  leveling  screws 


ADJUSTMENTS   OF   THE   LEVEL. 


65 


and  the  slow-motion  screw,  and  carefully  turn  the  telescope  in 
the  Y's  to  its  former  position.  If  the 
intersection  of  the  wires  now  remains 
on  the  point,  the  adjustment  is  cor- 
rect for  the  distance  at  which  the 
point  is,  and  this  is  all  that  is  usually 
given  for  this  adjustment. 


45.  Adjustment  of  objective  slide. 
It  is  also  correct  for  all  distances  if 
the  object  slide  is  in  adjustment ; 
that  is,  if  its  ring  is  so  centered  that 
the  optical  axis  of  the  objective  moves 
in  the  axis  of  the  telescope  tube. 
To  determine  whether  this  is  so,  test 
for  the  collimation  error  as  just  de- 
scribed, but  on  a  very  near  point.  If 
there  appears  to  be  an  error,  it  must 
be  corrected  by  adjusting  the  object 
slide.  To  do  this,  remove  the  ring 
that  covers  the  screws  controlling  the 
slide  ring,  and  adjust  with  these 
screws  so  that  one  half  the  error  ap- 
pears to  be  corrected.  Test  to  see 
whether  the  adjustment  is  correct,  and 
again  test  on  the  distant  point,  and,  if 
necessary,  adjust  the  wire,  and  repeat 
till  the  adjustments  are  both  correct. 
The  correctness  of  this  method  is 
shown  by  Fig.  35.  The  figure  is 
much  distorted. 

Let  o  be  the  optical  center  of  the 
objective.  It  is  assumed  that  the 
maker  has  so  placed  it  in  its  ring  that 
it  is  practically  in  the  axis  of  the 
telescope  when  drawn  clear  back  as 
shown.  This  is  probably  not  exactly 
true,  but  the  error  is  less  than  can 
be  determined  by  the  surveyor.  Let  D  be  the  ring  that  carries 
R'M'D  SURV.  —  5 


66  LEVELING. 

the  object  slide.  It  is  adjusted  by  the  screws  SS.  It  is  shown 
much  out  of  the  center  of  the  tube.  Let  wl  be  a  wire  supposed 
not  to  be  in  the  center  of  its  tube,  that  is,  out  of  adjustment. 
The  outer  end  of  the  objective  tube  slides  in  the  fixed  ring  Gr. 
If  the  instrument  is  perfectly  made,  it  is  evident  that,  if  the 
objective  could  be  brought  to  the  plane  of  this  ring  6r,  the 
optical  center  would  be  in  the  axis  of  the  telescope  tube.  When 
the  glass  is  focused  on  a  very  distant  object,  the  objective  ap- 
proximates to  the  plane  of  the  ring,  and  hence  the  optical  cen- 
ter is  nearly  in  the  axis  of  the  tube,  but  not  necessarily  exactly 
so.  In  the  case  presented,  it  could  not  be  just  in  the  axis. 

If  the  telescope  is  directed  to  a  distant  object,  as  a  rod 
jR,  the  point  a  of  the  rod,  that  will  appear  to  be  covered  by  the 
intersection  of  the  wires,  will  be  found  in  the  line  joining  the 
intersection  of  the  wires  with  the  optical  center  of  the  objec- 
tive. If  the  telescope  is  turned  upside  down,  the  wire  will 
move  to  Wy,  and  the  point  that  will  seem  to  be  covered  will 
be  the  point  6,  found  as  was  a.  If  now  the  optical  center  is  in 
the  axis  of  the  telescope,  and  the  wire  is  moved  to  the  position 
w  in  the  prolongation  of  the  line  joining  the  point  midway 
between  a  and  b  and  the  optical  center,  it  will  be  in  the  center 
of  the  tube  and  in  adjustment.  If  the  optical  center  is  not 
exactly  in  the  center  of  the  tube,  the  position  of  the  wire  when 
moved  to  read  halfway  between  a  and  b  will  not  be  the  center 
of  the  tube  but  will  be  very  near  it,  depending  on  the  error  of 
the  objective.  It  will  be  off  center  about  the  same  amount 
that  the  optical  center  is  off.  Let  it  be  assumed  that  the 
optical  center  is  right,  and  that  therefore  the  wire  is  now 
right.  Make  the  telescope  again  normal. 

Let  the  glass  be  focused  on  a  near  object,  as  a  rod  at  Rr 
Let  it  be  supposed  that  in  throwing  the  glass  out  for  this 
purpose  o  moves  to  or  The  reading  on  the  rod  will  be  c, 
obtained  by  producing  the  line  wor  Invert  the  telescope,  and 
the  reading  will  be  d.  If  the  ring  D  is  now  moved,  the  object 
slide  will  swing  about  GrG-  as  a  pivot,  and  o2  will  be  moved 
toward  or  It  is  evident  that,  if  this  movement  is  enough  to 
make  the  wire  read  halfway  between  c  and  <#,  the  objective  is 
centered.  If  it  was  right  for  the  distant  rod,  it  may  now  be 
assumed  to  be  right  for  all  intermediate  distances,  since  it  is 


ADJUSTMENTS   OF   THE   LEVEL.  67 

supposed  that  the  maker  has  made  the  tube  true  in  form.  If 
this  is  doubted,  it  may  be  tested  in  the  same  manner  for  any 
distance.  Since  the  objective  was  not  exactly  centered  for  the 
distant  rod,  the  wire  was  not  exactly  adjusted,  and  hence  if  the 
test  is  not  made  again  on  the  distant  rod,  a  slight  error  will  be 
discovered  which  is  to  be  corrected  by  moving  the  wire,  and 
the  test  must  again  be  made  on  the  near  rod,  which  will  also 
probably  show  some  error  of  objective  which  must  be  corrected. 
By  continual  approximations  the  work  is  completed. 

Because  the  wire  is  not  quite  centered  the  first  time,  the  work 
may  be  hastened  by  adjusting  the  slide  for  about  or  quite  the 
whole  error  instead  of  half  the  error.  Each  wire  is  adjusted 
separately,  and  the  objective  slide  first  in  one  direction  and 
then  in  the  other  at  right  angles  thereto.  It  is  well  to  note 
that  this  adjustment  of  the  objective  is  seldom  made  because  it 
is  seldom  necessary.  It  may  be  considered  to  be  necessary  only 
when  the  instrument  has  been  taken  apart  or  has  had  extraor- 
dinarily severe  usage.  When  it  is  not  necessary,  the  adjust- 
ment of  the  line  of  collimation  is  a  very  simple  matter.  In 
some  makes  of  instruments  the  object  slide  is  permanently 
adjusted,  and  hence  no  provision  is  made  for  its  adjustment 
by  the  engineer. 

46.  Bubble  adjustment.  The  second  adjustment  is  made  in 
either  of  two  ways.  The  first  depends  on  the  equality  of  di- 
ameter of  the  rings  of  the  telescope  tube  that  rests  in  the  Y's. 
Since  by  the  first  adjustment  the  line  of  collimation  has  been 
made  to  coincide  with  the  axis  of  the  tube,  it  is  parallel  to  the 
axis  of  the  Y's,  if  the  rings  mentioned  are  equal  in  diameter. 
If  now  the  attached  bubble  tube  is  made  parallel  to  the  axis 
of  the  Y's,  it  will  be  parallel  to  the  line  of  collimation.  To 
test  it,  set  up  the  instrument,  and,  having  leveled  over  both 
sets  of  screws,  unfasten  the  clips  and  level  more  carefully  over 
one  set.  Remove  the  telescope  very  carefully  from  the  Y's  and 
turn  it  end  for  end  and  replace.  If  the  bubble  returns  to  the 
center  of  its  tube,  the  tube  is  parallel  to  the  axis  of  the  Y's.  If  it 
does  not,  bring  it  halfway  to  the  center  by  the  adjusting  screws 
at  the  end  of  the  bubble  tube.  The  adjustment  is  now  com- 
plete if  the  instrument  has  not  been '  moved  in  making  at. 


68  LEVELING. 

' 
Test  it  by  releveling  and  turning  end  for  end  in  the  Y's,  as 

before.1 

47.  The  "peg"  method.  This  method  is  considered  better 
by  almost  all  engineers,  and  is  as  follows :  Set  the  instrument 
in  a  fairly  level  plot  of  ground  midway  between  two  stakes  that 
have  been  driven  firmly  into  the  ground,  from  two  hundred 
to  four  hundred  feet  apart.  Read  a  rod  on  each  stake.  (Re- 
member to  have  the  bubble  always  in  the  center  of  its  tube 
when  a  reading  is  taken.)  Since  the  rods  are  at  equal  dis- 
tances from  the  instrument,  errors  of  reading  due  to  errors 
of  adjustment  will  disappear  in  subtracting,  and  hence  the 
difference  in  readings  will  give  the  true  difference  in  elevation 
of  the  two  stakes.  The  instrument  is  now  removed  to  a  point 
in  line  with  the  two  stakes,  but  beyond  one  of  them.  Some 
engineers  take  the  instrument  beyond  one  stake  a  distance 
equal  to  one  tenth  of  the  distance  between  the  stakes.  Others 
set  the  instrument  at  one  of  the  stakes  so  that  the  eye  end  of 
the  instrument  in  swinging  around  will  just  clear  a  rod  held  on 
the  stake.  The  procedure,  adopting  the  first  method,  will  be 
explained,  and  the  student  may  work  out  the  process  for 
the  second  method.  It  will  be  supposed  that  the  instrument 
has  been  set  up  beyond  one  stake  a  distance  equal  to  one  tenth 
of  that  between  the  stakes,  and  with  one  diagonally  opposite 
pair  of  leveling  screws  in  line  with  the  stakes.  Any  other 
proportion  would  do  as  well. 

After  leveling,  with  the  telescope  pointing  in  the  line  of  the 
stakes,  a  reading  is  taken  on  the  near  stake.  If  there  were  no 
curvature  of  the  earth,  the  difference  in  level  of  the  two  stakes 
applied  to  the  reading  on  the  first  would  give  the  reading  that 
should  be  obtained  on  the  second,  provided  the  line  of  collima- 
tion  is  horizontal,  that  is,  parallel  to  the  axis  of  the  bubble  tube. 
Since  there  is  the  earth's  curvature  to  consider,  even  if  the 
instrument  is  in  adjustment,  the  rod  at  the  distant  stake  will 
read  more  than  above  noted,  by  the  amount  of  the  earth's 
curvature,  less  the  effect  of  refraction.2  Therefore,  to  obtain 
the  proper  reading  for  the  distant  rod  there  must  be  added 

1  The  principle  on  which  this  adjustment  depends  is  stated  in  Art.  28. 

2  See  Art.  40. 


ADJUSTMENTS  OF   THE   LEVEL.  69 

to  the  reading  on  the  near  rod,  the  difference  of  elevation  plus 
the  correction  for  curvature  for  the  given  distance. 

The  target  is  then  set  at  this  reading  and  the  rod  is  held  on 
the  distant  stake.  If  the  level,  when  sighted  to  the  rod,  reads 
above  the  target,  the  line  of  collimation  is  not  parallel  to  the 
bubble  tube,  but  points  up.  If  it  reads  below  the  target,  it  points 
down.  The  amount  of  error  in  reading  is  the  amount  that  the 
line  of  collimation  points  up  or  down  in  the  distance  between  the 
stakes.  In  the  distance  from  the  level  to  the  distant  stake,  it 
points  up  or  dpwn  eleven  tenths  of  this  apparent  error.  The 
proportion  will  vary  with  the  varying  proportions  of  the  dis- 
tances from  stake  to  stake  and  stake  to  instrument.  Hence  to 
get  a  horizontal  line  from  the  instrument  to  the  far  rod,  set  the 
target  in  the  line  of  collimation  and  then  raise  or  lower  it,  as 
the  case  may  be,  by  an  amount  equal  to  eleven  tenths  of  the 
apparent  error,  and  point  the  telescope  to  it  by  manipulating 
the  leveling  screws. 

The  line  of  collimation  now  being  horizontal,  the  bubble  tube 
will  be  parallel  to  it  if  the  tube  is  adjusted  with  reference  to  the 
telescope  so  that  the  bubble  comes  to  the  center.  This  is  done, 
as  explained  in  the  first  method,  by  the  screws  at  one  end  of  the 
tube.  The  test  should  be  again  made  to  see  that  no  movement 
has  taken  place  during  the  operation.  It  is  sometimes  stated 
that  the  adjustment  may  be  made  by  moving  the  bubble  or  the 
wire.  This  is  not  correct,  as  will  be  evident  from  the  preceding 
discussion  of  collimation  adjustment.  If  the  objective  is  perma- 
nently adjusted  by  the  maker,  the  second  adjustment  by  revers- 
ing in  the  Y's  may  be  made  first,  and  the  peg  method  may  follow, 
adjusting  the  wire. 

48.  Lateral  adjustment.  In  many  old  levels,  the  telescope 
is  free  to  turn  in  the  Y's.  If  this  is  the  case,  and  the  axis  of 
the  bubble  tube  is  not  in  the  vertical  plane  through  the  line 
of  collimation,  and  the  telescope  tube  becomes  turned  a  little, 
the  line  of  collimation  will  not  be  horizontal  when  the  bubble 
is  in  the  center  of  its  tube.  To  test  this,  level  the  instrument 
and  turn  the  telescope  slightly  in  the  Y's.  If  the  bubble  re- 
mains in  the  center  of  its  tube,  the  adjustment  is  perfect,  and 
nothing  need  be  done.  If  not,  adjust  the  tube  laterally  by  the 


70  LEVELING. 

lateral  adjusting  screws  at  one  end  of  the  tube.  Most  levels 
have  a  little  pin  in  one  of  the  clips  that  fits  in  a  notch  cut  in 
a  ring  on  the  telescope  and  prevents  the  telescope  from  turning 
in  the  Y's.  Nevertheless,  it  is  well  to  know  that  the  adjustment 
is  not  much  out. 

49.  Y  Adjustment.    The  third  adjustment  depends  on  Prin- 
ciple II.  stated  in  Art.  28.     It  is  performed  as  follows  :  The  test 
consists  in  carefully  leveling  the  instrument  over  both  sets  of 
screws  and  more  carefully  over  one  set,  and  the.n  swinging  the 
telescope  around  180°  on  the  vertical  axis  and  noting  whether  the 
bubble  remains  in  the  center  of  the  tube.     If  not,  the  axis  of 
the  instrument  is  not  vertical,  and  the  axis  of  the  Y's  is  not  per- 
pendicular to  it.     To  correct  the  error,  that  is,  to  make  the  axis 
of  the  Y's  perpendicular  to  the  vertical  axis  of  the  instrument, 
move  one  of  the  Y's  up  or  down  (as  may  appear  necessary  by 
the  position  assumed  by  the  bubble)  by  manipulating  the  large 
capstan  nuts  that  hold  the  Y  to  the  bar,  till  the  bubble  has 
moved  halfway  back  to  the  center  of  its  tube.     The  axis  of 
the  Y's  is  then  perpendicular  to  the  vertical  axis  of  the  instru- 
ment, provided  no  movement  has  taken  place.     To  test,  relevel 
and  reverse  the  telescope  and  make  any  further  correction  neces- 
sarj',  and  test  again  until  the  work  is  complete. 

In  making  any  of  the  adjustments  it  is  practically  impos- 
sible to  manipulate  the  adjusting  screws  without  to  some  extent 
disturbing  the  instrument.  Therefore,  an  adjustment  is  rarely 
made  perfectly  the  first  time,  and  must  always  be  tested  and 
repeated  till  complete.  If  an  instrument  is  badly  out  of  adjust- 
ment, it  will  be  advisable  first  to  adjust  it  approximately  all 
round,  and  then  readjust  more  carefully. 

50.  To  adjust  the  dumpy  level.     Reference  to  the  cut  of 
the  dumpy  level  will  show  that  the  axis  of  the  telescope  is 
made  by  the  maker  perpendicular  to  the  vertical  axis  of  revo- 
lution.    It  is  not  supposed  that  this  condition  will  change. 
It  will  also  be  noticed  in  the  cut  that  there  are  no  adjusting 
screws  for  the  object  slide.     It  is  permanently  adjusted.     This 
is  true  of  all  the  engineering  instruments  made  by  the  maker 
of  this  level.      Therefore,   since  the  axis  of  the  telescope  is 


ADJUSTMENTS   OF   THE   LEVEL.  71 

perpendicular  to  the  axis  of  revolution  and  the  optical  axis  of 
the  objective  is  permanently  adjusted,  if  the  axis  of  the  bubble 
tube  is  made  perpendicular  to  the  axis  of  revolution,  and  the 
line  of  collimation  is  made  parallel  to  the  bubble  axis,  the 
instrument  will  be  correctly  adjusted.  In  this  instrument, 
then,  the  order  of  adjustment  is  reversed. 

First,  the  bubble  tube  is  adjusted  to  be  perpendicular  to  the 
axis  of  revolution.  The  test  for  this  condition  is  made  as  in 
the  test  for  the  Y  adjustment  of  the  Y  level.  If  adjustment  is 
needed,  it  is  made  with  the  screws  at  the  end  of  the  bubble  tube. 
One  half  the  apparent  error  is  corrected. 

Second,  the  line  of  collimation  is  made 
parallel  to  the  axis  of  the  bubble  tube.  The 
test  for  this  condition  is  made  by  the  "peg" 
method,  as  already  described.  The  adjustment 
is  made  by  moving  the  wires  instead  of  the 
bubble. 

The   dumpy  level,   and   in   fact  all   levels, 
should  be  so   made    that  the  vertical  wire   is         FlG 
not  adjustable.     A  form  of  ring  meeting  this 
requirement  is  shown  in  Fig.  36,  taken  from  a  valuable  English 
work  on  Engineers'  Surveying  Instruments.1 

51.  To  adjust  any  level  on  a  metal  base.  To  adjust  in  this 
way,  say,  a  carpenter's  level,  or  a  striding  level  for  engineering 
instruments,  place  the  level  on  a  plane  surface  and  mark  its 
position,  and  note  the  position  of  the  bubble  in  its  tube. 
Reverse  the  level,  end  for  end,  and  place  it  in  the  same  posi- 
tion on  the  plane  surface.  If  the  bubble  is  as  far  from  the 
center  in  the  reversed  position  as  it  was  in  the  first,  the  axis 
is  parallel  to  its  base,  which  means  that  it  is  in  adjustment. 
If  not,  correct  one  half  the  difference  in  readings  by  what- 
ever screws  may  be  provided  for  the  adjustment  of  the  bubble. 
If  convenient,  the  bubble  may  be  brought  to  the  center  in  the 
first  position  by  tilting  the  plane  surface ;  then  when  the  level 
is  reversed,  the  movement  of  the  bubble  from  the  center  will 
correspond  to  twice  the  error  of  adjustment,  which  may  then  be 
corrected. 

1  Stanley's  "  Surveying  and  Leveling  Instruments." 


72 


LEVELING. 


MINOR  INSTRUMENTS. 

52.  Locke  hand  level.     This  is  a  small  instrument  for  quick 
approximations  where  no  great  degree  of  precision  is  required. 
It  is  shown  in  Fig.  37. 

It  consists  of  a  small  bubble  mounted  on  a  brass  tube  over 
a  hole  in  the  tube,  beneath  which,  and  within  the  tube,  is  a 
prism  occupying  one  half  of  the  tube.  Across  the  hole  is 

stretched  a  wire. 
There  is  usually  an 
eyepiece  of  low 
magnifying  power. 
The  eye,  in  looking 
through  the  tube, 
will  see  the  wire  and  the  bubble  reflected  by  the  prism  ;  and 
when,  if  the  instrument  is  in  adjustment,  the  bubble  appears 
to  be  bisected  by  the  wire,  the  line  of  vision  past  the  wire  is 
horizontal.  In  use,  the  leveler  and  hand  level  constitute  the 
leveling  instrument,  and  between  plus  and  minus  readings 
the  observer  must  remain  stationary.  There  are  other  forms 
of  the  Locke  level  than  that  shown  in  the  figure.  The  prin- 
ciple is  the  same  in  all. 

53.  Abney  level  and  clinometer.     This  is  a  combination  of 
a   hand   level   and   a   slope-measuring   instrument.      Such   an 
instrument  is  shown  in  Fi 


FIG.  37. 


FIG.  38. 


It  is  essentially  a  hand  level,  in  which  the  bubble  is  at- 
tached to  an  arm  that  carries  a  vernier,  and  may  be  moved 
over  a  small  arc.  To  determine  a  slope  or  vertical  angle, 


MINOR   INSTRUMENTS. 


73 


bring  the  line  of  sight  to  the  slope,  and  turn  the  vernier  arm 
till  the  bubble  is  seen  on  the  wire.  The  bubble  axis  will  then 
be  horizontal,  and  the  angle  between  it  and  the  line  of  collima- 
tion  will  be  indicated  by  the  vernier  reading. 

54.  Gurley's  monocular  and  binocular  hand  levels.  These 
are  telescopic  hand  levels  by  which  readings  are  more  defi- 
nitely determined  on  a  rod  at  some  distance  than  is  possible 


FIG.  39. 


with  the  ordinary  hand  level  which  is  not  a  telescope.  The 
Gurley  levels  are  shown  in  Fig.  39.  The  author  thinks  the 
monocular  hand  level  the  best  yet  devised. 

55.  Adjustment  of  the  Locke  and  Abney  levels.  For  the 
Locke  level,  get  by  some  means  a  horizontal  line,  as  a  sheet  of 
water  of  limited  extent ;  if  the  latter  is  not  at  hand  use  a  level- 
ing instrument,  or  indeed  the  hand  level  itself,  held  midway 
between  two  rods  or  trees  some  distance  apart.  Having  ob- 
tained the  horizontal  line,  stand  with  the  hand  level  at  one 
end  of  it,  and,  with  the  eye  and  wire  in  the  horizontal  line, 
note  whether  the  bubble  appears  to  be  on  the  wire.  If  it  does 
not,  adjust  by  moving  the  frame  carrying  the  wire  by  means 
of  the  little  screws  at  the  end  of  the  bubble  mounting. 

To  adjust  the  Abney  level  and  clinometer,  set  the  vernier 
to  read  zero,  and  test  as  for  the  hand  level;  adjust  the  bubble 
by  screws  provided  for  that  purpose,  or  adjust  the  wire  by  a 
slide  arrangement  provided  in  some  instruments. 


74 


LEVELING. 


LEVELING  WITH   THE   BAROMETER. 

56.  Barometer  described.      Very   rough  leveling   work   is 
sometimes  done  with  the  aneroid  barometer.     The  barometer 
is  an  instrument  for  measuring  the  pressure  of  the  atmosphere. 
The  only  reliable  form  is  the  mercurial  barometer.     This  is 

not  a  convenient  portable  in- 
strument. The  aneroid  barom- 
eter is  an  attempt  at  portable- 
ness.  It  is  made  in  various 
forms,  the  more  common  of 
which  is  shown  in  Fig.  40. 

It  consists  essentially  of  a 
metallic  box,  with  a  thin  corru- 
gated top,  from  which  the  air  is 
exhausted.  With  the  varying 
pressure  of  the  atm.osphere,  the 
top  of  the  box  rises  and  falls. 
Its  motion  is  communicated  by 
levers  and  chains  to  a  pointer 
moving  around  a  dial,  gradu- 
ated by  comparing  the  movement  of  the  pointer  with  that  of 
a  mercurial  barometer.  Hence  the  "inches"  on  the  dial  are 
not  true  inches  nor  of  uniform  length. 

57.  Theory.      The  use  of  the  barometer  to  determine  dif- 
ferences of  altitude  depends  on  the  supposition  that  the  atmos- 
phere is  composed  of   a  series  of  layers  of  air  of  uniformly 
upward  decreasing  density.     If  this  were  true,  the  pressure 
indicated  by  the  barometer  would  grow  uniformly  less  as  the 
barometer  were  carried  to  higher  altitudes,  and  it  would  be  easy 
to  establish  by  experiment  a  relation  between  the  altitude  and 
the  barometric  reading.     The  supposition  is  not  entirely  cor- 
rect.     The   pressure  of   the  atmosphere  at  any  one  place  is 
affected  by  humidity,  and  very  considerably  by  temperature. 
Moreover,   owing  to  the  movement  of    the    atmosphere,   two 
distant  places  of  equal  altitude,  having  at  a  given  time  equal 
temperatures,  will  not  have  at  the  same  time  equal  barometer 
readings. 


FIG.  40. 


THE  BAROMETER.  75 

It  is  therefore  difficult  to  establish  a  relation  between  the 
altitude  ^  and  the  barometer  readings.  It  has  been  attempted 
by  many  persons,  with  the  result  that  there  are  many  formulas 
for  determining  the  differences  in  altitude  from  the  readings  of 
the  barometer.  These  formulas  do  not  differ  so  much  in  method 
of  deduction  as  in  the  experimental  constants  used.  Perhaps 
one  of  the  best  is  that  developed  by  Mr.  William  Ferrall.1 

Modified  by  omitting  the  terms  for  correcting  for  humidity 
and  latitude,  which  are  small  in  effect,  and  one  or  two  other 
still  smaller  terms,  this  formula  is 

ff=  60521.5  log  -|-f  1  +  0.001017  (t  +  t1  -  64°)  \ 
-»i\ 

in  which  II  is  the  difference  in  altitude  of  two  points  where 
observations  are  taken,  B  and  S1  are  the  barometer  readings  at 
the  lower  and  higher  points  respectively,  and  t  and  t'  are  the 
Fahrenheit  temperatures  at  the  two  points. 

58.  Barometric  tables.  It  is  convenient  in  tabulating  this 
formula  for  practical  use  to  consider  that  t  +  t'  is  100°,  and 
that  H  is  the  difference  in  altitude  between  two  points  whose 
altitudes  above  a  given  datum  are  to  be  directly  determined 
from  the  tables.  Thus  if  t  +  t'  =  100°,  and  the  given  datum  is 
such  that  the  barometer  reading  is  30  inches,  the  formula  is 

/  Qft  30 

H=  60521.5  (  1  +  .001017  x  36  )  (log  f£-  -  log  ^ 

V        Bl  B 

=  62737  log  ^  -  62737  log^- 
±f1  B 

30 

Tables  may  be  constructed  for  the.  quantity  62737  log  —  , 

B 

with  B  as  the  argument,  and  such  a  table  would  be  entered 
twice,  once  for  each  barometer  reading,  and  the  difference  of 
the  tabular  values  found  would  be  the  value  of  H  if  £+£'  =  100° 
and  there  is  no  correction  for  humidity.  Table  III.,  page  362, 
is  such  a  table.  For  other  than  a  mean  temperature  of  50°,  and 
for  average  hygrometric  conditions,  a  correction  is  to  be  applied 
to  the  result.  This  correction  is  a  linear  function  of  the  value 
of  H  'already  found  such  that 


See  Appendix,  10,  "  United  States  Coast  and  Geodetic  Survey  Report  "  for  1881. 


76  LEVELING. 

in  which  Al  and  A%  are  the  values  from  Table  III.,  and  C  is  the 
temperature  and  humidity  correction  coefficient.  Values  of  C 
are  given  in  Table  II.,  page  361. 

59.  Practical  suggestions.      The  aneroid  barometer  should 
be  frequently  compared  with  a  mercurial  barometer,  and  the 
errors  of  reading  at  different  parts  of  the  scale  should  be  noted. 
This  can  be  well  done  only  under  the  bell  of  an  air  pump. 

The  scale  of  altitudes  on  some  barometers  is  useless. 

Some  barometers  are  marked  "  compensated,"  meaning  that 
no  temperature  corrections  need  be  applied.  This  is  a  mistake. 
The  temperature  corrections  should  always  be  applied. 

Always  read  the  barometer  when  it  is  horizontal.  Carry  it 
in  a  strong  case,  and  do  not  let  the  warmth  of  the  body  affect 
it.  Do  not  use  it  for  altitudes  just  before  or  after,  or  during 
a  storm. 

In  determining  the  difference  of  altitude  of  two  widely 
separated  points,  stop  once  or  twice  for  a  half  hour  or  so  on 
the  journey  between  the  points,  note  the  reading  of  the  ba- 
rometer at  the  beginning  and  end  of  the  stop,  and  thus  try 
to  determine  whether  the  atmospheric  conditions  are  changing 
and,  if  so,  at  what  rate.  A  correction  can  thus  be  determined 
for  the  first  reading,  on  the  supposition  that  the  observed  change 
has  taken  place  also  at  the  starting  point.  Tap  the  box  gently 
before  reading. 

60.  Accuracy  of  the  method.      If  barometer  readings   are 
taken  simultaneously  daily  at  two  widely  separated  stations  for 
from  one  to  six  years,  the  utmost  that  can  be  expected  is  an 
error  of  about  one  half   of   one  per  cent,  though  occasional 
results  are  much  closer.    Single  observations  taken  on  the  same 
day  with  the  same  or  compared  barometers  and  at  distances  of 
a  few  hours'  walk  will  give  results  with  errors  of  from  one  per 
cent  to  three  per  cent  and  more.     An  error  in  temperature  of 
five  degrees  will  cause  an  error  in  result  of  one  per  cent  or 
more.     Single  observations  taken  a  day  or   more  apart  and 
at  stations  a  day's  journey  or  more  apart  can  not  be  depended 
on  to  give  results  much  closer  than  the  nearest  ten  per  cent, 
though  they  will  frequently  do  better  than  this  and  sometimes 
worse. 


CHAPTER   IV. 

DETERMINATION  OF  DIRECTION  AND  MEASUREMENT  OF 
ANGLES. 

61.  Instruments   mentioned.     In   almost   all   surveys   it   is 
necessary  to  measure  distances  and  angles. 

Angles  are  measured  by  means  of  one  of  several  instru- 
ments, according  to  the  character  of  the  work.  The  most 
commonly  used  are  the  compass  and  the  transit. 

The  compass  is  an  instrument  for  determining  directions 
and,  indirectly,  angles  ;  while  the  transit  is  an  instrument  for 
determining  angles  and,  indirectly,  directions. 

Almost  all  of  the  old  land  surveys  of  this  country  have 
been  made  with  the  compass  as  the  angle-measuring  instru- 
ment ;  and  while  this  instrument  has  been,  to  a  great  extent, 
supplanted  by  the  transit,  there  are  still  a  great  many  com- 
passes sold  annually.  As  a  discussion  of  the  compass  will 
make  clearer  some  surveying  methods,  it  will  be  here  described. 
Mention  mast  be  made  of  the  sextant,  which  is  an  angle-meas- 
uring instrument  whose  use  in  surveying  is  confined  to  certain 
particular  operations,  as  in  exploratory  surveying  or  in  the 
location  of  soundings  taken  off  shore.  A  description  of  this 
instrument  is  given  on  pages  294-298. 

Mention  should  also  be  made  of  the  solar  compass  and 
transit,  instruments  for  determining  the  true  meridian  by  an 
observation  on  the  sun.  The  solar  transit  is  described  on 
pages  116-126. 

THE  COMPASS. 

62.  Description.     The  compass  consists  of  a  line  of  sight 
attached  to  a  graduated  circular  box,  in  the  center  of  which 
is  hung,  on  a  pivot,  a  magnetic  needle. 

77 


78  DIRECTION  AND  ANGLES. 

At  any  place  on  the  earth's  surface^  the  needle,  if  allowed 
to  swing  freely,  will  assume  a  position  in  what  is  called  the 
magnetic  meridian  of  the  place.     If,  then,  the  direction  of  any 
line  is  required,  the  compass  may  be  placed  at  orie  end  of  the 
line  and  the  line  of  sight  may  be  made  to  coincide  with  the  line. 
The  needle  lying  in  the  magnetic  meridian,  and  the  zero  of 
the  graduations  of  the  circular  needle  box  being  in  the  line  of 
sight,  the  angle  that  the  line  on  the  ground  makes  with  the 
magnetic  meridian  is  read  on   the   graduated  circle.     If  the 
magnetic  meridian  coincided   everywhere   with  the  true 
meridian,  or  even  if  the  angle  between  them  were  con- 
stant, the  compass  would  be  a  far  more  valuable  instru- 
ment than  it  is.     The  compass,  however,  has  its  field,  and 
in  that  field  is  a  very  valuable  tool.     A  very  good  form 
of  compass  is  shown  in  Fig.  41. 

It  is  sometimes  set  on  a  tripod  head  and  sometimes 
on  a  Jacob  staff,  which  is  a  single-pointed  staff  with  a 


FIG.  41. 

head  fitted  to  receive  the  compass.  The  sights  SS  are  attached 
to  the  main  plate  of  the  instrument  by  the  screws  seen  below  the 
plate.  The  level  vials  encased  in  brass  tubes  are  fastened  to  the 
plate  by  screws  passing  through  the  plate  from  below,  by  which 
screws  the  bubbles  may  be  adjusted  parallel  to  the  plate.  The 
needle  Crests  on  a  steel  pivot  that  is  screwed  into  the  plate, 
and  the  needle  is  lifted  from  the  pivot,  when  not  in  use,  by  the 
ring  R  operated  by  a  lever,  which  is  in  turn  moved  by  the 
screw  whose  head  is  seen  just  below  the  plate  directly  in  front. 
On  the  left  is  a  marker  M  used  to  keep  track  of  the  chaining. 
Next  to  this  is  an  arc  A  and  a  vernier,  moved  by  the  tangent 
screw  V.  The  arc  is  attached  to  the  plate,  and  the  vernier  to 


COMPASS   ADJUSTMENTS.  79 

an  arm  that  is  fastened. to  the  compass  box,  which  box  may  be 
thus  turned  through  a  small  angle  by  the  tangent  screw.  This 
vernier  is  called  a  declination  vernier.  Its  use  will  appear  later. 
The  compass  box  is  graduated  from  the  north  and  south  points 
in  each  direction  90°.  When  the  zero  of  the  vernier  coincides 
with  the  zero  of  the  arc  -A,  the  line  of  zeros  of  the  compass  box 
is  coincident  with  the  line  of  sight.  The  spring  catch  0  holds 
the  compass  on  the  tripod  head,  and  the  clamp  K,  acting  on  a 
concealed  spring  that  encircles  the  tripod  head,  serves  to  clamp 
the  instrument  so  that  it  will  not  revolve  about  the  vertical 
axis.  There  is  a  small  coil  of  wire  seen  on  one  end  of  the 
needle.  This  counteracts  the  magnetic  force  that  would  other- 
wise cause  the  needle  to  dip.  This  means  that  the  direction  of 
magnetic  force  or  magnetic  pull  is  not  horizontal,  but  is  inclined, 
and  if  the  needle  is  free  to  dip,  it  will  lie  in  the  direction  of 
magnetic  pull.  This  vertical  component  of  magnetic  force  that 
causes  the  dip,  varies  in  different  places,  and  the  little  coil  of 
wire  may  be  moved  nearer  to,  or  further  from,  the  pivot  to  allow 
for  this  variation. 

Since  only  horizontal  angles  are  measured  with  the  compass 
and  used  in  land  surveying,  it  is  necessary  to  have  the  plates 
horizontal,  and  the  sights  vertical  when  working.  This  is 
accomplished  by  means  of  a  ball  and  socket  joint  attached  to  the 
tripod  head,  and  the  bubbles  on  the  plate.  The  plate  having 
been  leveled,  the  joint  is  clamped  by  simply  screwing  tight  the 
milled  head  by  which  the  spindle  is  attached  to  the  head  of  the 
tripod. 

COMPASS  ADJUSTMENTS. 

63.  Requirements  mentioned.     To  determine  properly  the 
direction  of  lines  with  the  compass,  it  is  necessary  (1)  that  the 
plate  be  level,  the  sights  vertical  with  their  line  passing  through 
the  zeros  of  the  graduated  box  when  the  zeros  of  the  vernier 
and  arc  A  coincide  ;   (2)  that  the  needle  be  straight ;   (3)  that 
the  pivot  be  in  the  center  of  the  circle. 

64.  Plate  bubbles.     The  instrument  is  readily  leveled  if  the 
bubbles  are  parallel  to  the  plate  and  the  plate  perpendicular  to 
the  vertical  axis.     Test  the  parallelism  of  the  bubbles  and  plate 


80  DIRECTION  AND  ANGLES. 

by  the  method  of  Art.  28,  and  correct,  if  necessary,  by  the 
screws  already  mentioned.  In  doing  this  the  plate  is  reversed 
on  the  vertical  axis.  It  is  assumed  that  the  plate  is  perpendicu- 
lar to  the  vertical  axis.  If  not,  it  must  be  made  so  by  the 
maker. 

65.  Sights.     To  test  the   perpendicularity   of   the    sights, 
turn  the  leveled  instrument  on  a  suspended  plumb  line  and  see 
whether  the  line  traverses  the  slits  in  the  standards,  turning 
each  end  separately  to  the  plumb  line.     If  the  sights  are  found 
to  be  out  of  plumb,  they  may  be  adjusted  by  removing  them  and 
filing  the  bottoms  till  they  are  found  to  be  correct.      With 
proper  care  they  should  not  get  out,  and  this  is  not  a  common 
error.     Strips  of  paper  may  be  inserted  under  one  edge,  if  the 
surveyor  prefers  this  to  filing. 

To  determine  whether  the  plane  of  the  sights  includes  the 
line  of  the  zeros  when  the  vernier  is  set  to  read  zero,  stretch  two 
fine  threads  through  the  slits  in  the  sights  and,  looking  down, 
see  whether  the  plane  of  the  threads  includes  the  line  of  zeros  of 
the  graduated  box.  If  not,  the  error  should  be  corrected  by  the 
maker.  The  error  may  be  allowed  for,  however,  by  moving  the 
vernier  till  the  zeros  are  in  the  plane  of  the  threads,  or  parallel 
to  it,  noting  the  angle  that  has  been  turned,  and  seeing  that 
thereafter  the  vernier  is  set  at  this  angle  when  working.  If 
the  line  of  sights  fails  to  pass  through  the  center  by  one  tenth 
of  an  inch,  the  error  in  the  direction  of  a  line  ten  feet  long  will 
be  about  three  minutes,  and  will  be  less  as  the  line  is  longer. 

66.  Needle  and  pivot.     If  the  needle  is   straight  and  the 
pivot  is  in  the  center,  the  two  ends  will  read  180°  apart.     If 
both  of  these  conditions  are  not  fulfilled,  the  two  ends  will 
read  other  than  180°  apart.      If  it  is  observed   that  one  of 
these  faults   exists,   the   instrument   should  first  be  tested  to 
determine  which  one.     This  is  done  by  turning  the  plates  on 
the  vertical  axis  and  noting  whether  the  difference  in  the  two 
end  readings  remains  constant:  if  so,  the  needle  is  bent;  if  not, 
the  pivot  is  not  in  the  center,  and  the  needle  may  or  may  not 
be  bent.     If  the  needle  is  found  to  be  bent,  remove  the  glass 
cover  from  the  box  by  unscrewing  it,  take  the  needle  from  the 


COMPASS   ADJUSTMENTS. 


81 


pivot,  and  carefully  bend  it  till  straight.  If  the  pivot  is  found 
to  be  out  of  center,  remove  the  needle  and  carefully  bend  the 
top  of  the  pivot  over  until 
it  is  in  the  center.  To 
determine  which  way  to 
bend  it,  turn  the  box  till 
that  position  of  the  box 
is  found  that  gives  the 
greatest  error  in  end  read- 
ings of  the  needle,  and 
bend  the  pivot  at  right 
angles  to  the  position  of 
the  needle.1 

If  there  is  reason  to 
think  that  both  errors  are 
involved,  turn  the  instru- 
ment as  before  to  get  the 
maximum  error  of  end  readings,  and  read  the  angle  greater 
than  180°  between  the  ends,  calling  this  R.  Now  revolve  the 
instrument  180°,  in  the  way  described  below,  and  read  the 
angle  on  the  same  side  of  the  needle,  calling  this  reading  R' . 
Let  p  be  half  of  the  error  due  to  eccentricity  of  pivot,  and  n 
be  half  of  the  error  due  to  bent  needle;  then,  from  Fig.  42,  it 
is  seen  that 

180°+(2j9  +  2w)=  R-,  (1) 

180°  -  (2p  -  2  n)  =  R'.  (2) 

Solving  for  p  by  subtraction,  and  for  n  by  addition,  there  results 
R-R' 


FIG.  42. 


P  = 


R+R'  -  360 


Bend  the  needle  so  that  each  end  moves  over  n  degrees.  Then 
bend  the  pivot  at  right  angles  to  the  needle,  toward  the  center, 
till  the  needle  ends  read  180°  apart. 

Otherwise,  turn  the  box  till  the  needle  ends  read  180°  apart. 
Turn  the  box  180°,  and  if  the  two  ends  do  not  read  opposite 

1  The  student  should  show  the  correctness  of  these  statements  by  a  diagram. 
K'M'D  SURV.  — 6 


82  DIRECTION   AND   ANGLES. 

divisions,  bend  the  ends  through  half  the  difference  between 
the  first  and  second  positions.  The  needle  is  now  straight,  and 
the  pivot  may  be  tested  and  corrected  as  above  described. 

The  instrument  may  be  revolved  180°  by  noting  an  object 
to  which  it  points  when  first  set  and  then  revolving  till  the 
sights  again  point  to  the  same  object. 

If  the  needle  is  sluggish  in  its  movement,  it  may  be  due  to 
one  or  both  of  two  causes.  The  pivot  may  be  blunt,  or  the 
needle  may  have  lost  some  of  its  magnetic  strength.  Take  the 
needle  off  the  pivot  and  examine  the  pivot.  If  it  is  found 
blunt,  unscrew  it  and  carefully  grind  it  down  on  a  fine  oilstone. 
If  the  pivot  seems  perfect,  the  trouble  is  with  the  needle.  Lay 
the  needle  down  and  rub  each  end  from  the  center  out  with 
that  end  of  a  bar  magnet  which  attracts  the  end  of  the  needle 
that  is  being  rubbed.  In  passing  the  magnet  back  from  end  to 
center,  raise  it  some  distance  above  the  needle ;  otherwise  the 
backward  movement  will  tend  to  counteract  the  rubbing.  The 
needle  may  also  be  remagnetized  by  placing  it  in  the  magnetic 
field  of  a  strong  electro-magnet,  as  a  dynamo.  If  placed  on  the 
magnet  in  a  wrong  direction,  the  south  end  will,  when  replaced, 
point  north.  To  correct  this,  replace  the  needle  on  the  magnet 
in  the  opposite  direction. 

Two  needles  at  the  same  place  may  not  point  in  the  same 
direction.  This  will  probably  be  due  to  the  fact  that  their 
magnetic  axes  do  not  coincide  with  their  geometric  axes.  This 
seems  to  be  unavoidable.  For  this  reason  the  needle  should  be 
narrow  and  deep  rather  than  flat  and  wide.  There  is  a  differ- 
ence of  opinion  about  this,  based  on  difference  of  manufacturing 
processes. 

The  angle  between  the  true  meridian  and  the  magnetic 
meridian  should  be  determined  by  the  instrument  that  is  to  be 
used  in  a  given  piece  of  work. 

The  glass  cover  of  the  compass  box  may  become  electrified 
by  rubbing,  and  attract  the  needle.  This  is  remedied  by  touch- 
ing the  glass  with  the  moistened  finger. 

When  the  compass  is  being  moved,  the  needle  should  always 
be  lifted  from  the  pivot.  When  put  away,  the  needle  ^should 
be  allowed  to  swing  to  the  magnetic  meridian  and  then  lifted. 
It  retains  its  magnetic  strength  longer  in  the  meridian. 


USE   OF   THE   COMPASS. 


USE  OP  THE  COMPASS. 

67.  Bearing.     The  use  of  the  compass  is  to  determine  the 
directions,  or,  technically,  the  bearings  of  lines ;  or  to  mark  out 
lines  whose  bearings  are  given. 

The  bearing  of  a  line  is  the  horizontal  angle  between  a  ver- 
tical plane  including  the  line,  and  the  meridian  plane  through 
one  end  of  the  line,  and  is  measured  from  the  north  or  south 
points  90°  each  way.  Thus  the  bearing  of  a  line  extending  in  a 
direction  midway  between  the  north  and  east,  would  be  N.  45° 
E.  or  S.  45°  W.,  according  to  the  end  from  which  the  bearing 
is  read.  This  gives  rise  to  two  terms,  forward  bearing  and 
lack  bearing,  the  forward  bearing  being  the  bearing  in  the 
direction  in  which  the  survey  is  being  run,  and  the  back  bear- 
ing being  that  read  in  the  reverse  direction.  These  two  bear- 
ings (omitting  the  effect  of  convergence  of  the  meridians  of 
the  two  ends  of  the  line,  which  is  an  inappreciable  amount  in 
an  ordinary  compass  survey)  should  be  numerically  equal,  but 
with  opposite  letters  to  express  the  direction.  The  bearings 
are  never  read,  east  so  many  degrees  south  or  north,  or  west  so 
many  degrees  north  or  south ;  but  always  from  the  north  or 
south  points. 

68.  To  determine  the  bearing  of  a  line.     Set  the  compass  over 
one  end,  level  the  plates,  lower  the  needle,  turn  the  north  end 
of  the  compass  box  toward  the  distant  end  of  the  line,  and 
bring  the  line  of  sights  carefully  in  the  direction  of  the  line. 
If  the  declination   vernier  reads  zero,  the  north  end  of  the 
needle  will  show  the  bearing  of  the  line. 

69.  To  lay  out  a  line  of  given  bearing.     Set  the  compass  over 
a  point  on  the  line,  level  the  plates,  drop  the  needle,  turn  the 
compass  on  its  vertical  axis  till  the  north  end  of  the  needle 
reads  the  given  bearing.     The  line  of  the  sights  is  now  in  the 
required  line,  which  may  be  ranged  out. 

70.  To  run  a  traverse.     This  is  to  determine  the  lengths  and 
bearings  of  a  series  of  connected  lines.     These  may  form  the 
sides  of  a  farm,  or  inclose  a  pond,  or  be  the  center  line  of  a 
crooked  road.     If  there  are  no  natural  objects  to  mark  the 
angle  points,  flags  are  set  at  those  points  by  a  flagman,  the 


84 


DIRECTION   AND   ANGLES. 


compass  is  set  at  each  point  in  succession,  and  the  back  bearing 
of  the  preceding  line  and  the  forward  bearing  of  the  line  ahead 
are  determined  while  the  lines  are  being  measured  by  the  chain- 
men.  The  back  bearings  are  read  as  a  check  on  the  forward 
bearings  and  to  detect  local  attraction,  as  will  be  hereafter  ex- 
plained. In  occasional  hasty  work  the  compass  will  be  set  only 
at  every  second  corner,  and  the  back  bearing  of  one  line  (after- 
ward changed  to  forward  bearing)  and  the  forward  bearing  of 
the  other  line  will  be  determined.  This  should  not  ordinarily 
be  done.  If  the  chainmen  have  no  point  ahead  to  chain  toward, 
they  may  be  kept  in  line  by  the  man  at  the  compass. 

It  is  frequently  necessary  to  produce  a  line  for  a  consider- 
able distance  through  woods  or  over  hills,  one  end  not  being 
visible  from  the  other.  If  the  bearing  of  the  line  is  known, 
turn  the  compass,  placed  on  one  end  of  the  line,  in  the  direc- 
tion of  the  line,  and  direct  the  chainmen  as  far  as  they  can  be 
well  seen.  Establish  a  point  by  driving  a  stake  in  the  line  at 
the  end  of  the  last  measurement,  or  by  merely  setting  the  flag 
in  the  ground,  to  be  pulled  out  Avhen  the  compass  is  brought 
up,  and  set  the  compass  over  the  established  point  and  continue 
the  line. 

If  a  tree  or  small  building  obstructs  the  line,  and  a  few 
inches  on  one  side  or  the  other  are  of  no  consequence,  as  is  the 
case  in  most  surveys  made  with  the  compass,  set  the  compass 
beyond  the  obstruction,  as  nearly  on  the  line  as  can  be  estimated, 
and,  turning  the  sights  in  the  direction  of  the  known  bearing, 
proceed  with  the  line.  Other  obstacles  are  discussed  on 
pages  201-204. 

71.  Notes.  The  notes  that  are  taken  may  be  kept  in  the 
following  form,  if  nothing  more  than  a  record  of  the  traverse 
is  required  : 


STATION. 

BEARING. 

DIST.  CHAINS. 

etc. 

etc. 

etc. 

D 

S.  72°  45'  E. 

10.40 

C 

S.  52°  00'  E. 

7.30 

B 

N.  23°  30'  E. 

6.74 

A 

N.  36°  15'  W. 

4.63 

USE   OF   THE   COMPASS.  85 

The  notes  read  up  the  page  so  that  the  line  may  appear  to  be 
ahead  as  one  walks  along  with  the  open  notebook.  Other 
notes  are  usually  required,  as  distances  to  points  passed,  fences 
crossed,  roads  and  streams  crossed,  distances  right  or  left  to 
objects  near  the  line,  as  will  be  hereafter  described.  In  such 
cases,  a  sketch  of  the  work  with  all  the  necessary  measurements 
marked  on  it  is  the  best  possible  form  of  notes  to  take  in  the 
field.  This  may  afterwards  be  converted  into  the  form  shown 
on  page  229.  The  name  and  date  of  survey  should  always  be 
recorded  on  every  page  of  the  notebook  used  and  the  names  of 
those  engaged  on  the  work  should  be  entered  at  least  once. 

72.  Angles.    To  determine  the  angle  formed  by  two  lines  at 
their  point  of  intersection,  set  the  compass  over  the  point  of 
intersection  and  read  the  bearing  of  each  line.    From  these  bear- 
ings determine  the  angle.     There  will  always  be  two  supplemen- 
tary angles.     If  the  lines  are  conceived  to  be  run  from  the  end 
of  one  to  the  point  of  intersection,  to  the  end  of  the  second, 
and  so  on,  as  in  running  a  traverse,  the  angle  between  the 
prolongation  of  line  one,  and  line  two,  measured  in  the  direc- 
tion in  which  the  line  is  conceived  to  bend,  is  called  the  deflec- 
tion angle,  or,  in  a  closed  survey,  the  exterior  angle.     Some 
examples  in  converting  bearings  to  angles,  and  the  reverse,  will 
be  found  in  the  problems  in  the  Appendix,  pages  324,  325. 

73.  Cautions.     It  is  better  to  keep  the  north  end  of  the 
compass  box  ahead,  and  to  read  the  north  end  of  the  needle. 
The   north   end  of  the   needle  is  distinguished  either  by  its 
peculiar  shape  or  by  the  fact  that  the  coil  of  wire  is  on  the 
other  end.     The  north  end  of  the  compass  box  is  distinguished 
by  a  peculiar  figure,  usually  a  conventional  fleur  de  lis.     If  the 
south  end  of  the  box  is  ahead,  read  the  south  end  of  the  needle. 
Care  should  be  exercised  in  the  use  of  the  compass,  both  in 
directing   the   sights    and    in    reading    the    needle;    and  the 
beginner    should    be    particularly    careful    to    read    from    the 
north  or  south  point  according  to  which  is  nearer  the  north 
end  of   the  needle.      Greater   differences   in   results   may   be 
expected  from  different  instruments  used  by  the  same  man, 
than  from  the  same  instrument  used  by  different  men,  if  the 
men  use  care  in  their  work. 


86  DIRECTION   AND   ANGLES. 

MAGNETIC  DECLINATION. 

74.  Declination  defined.     Thus  far  all  bearings  have  been 
assumed  to  be  taken  with  reference  to  the  magnetic  meridian. 
If  the  magnetic  meridian  and  the  true  meridian  were  coinci- 
dent at  all  places  on  the  earth's  surface,  or  if  at  one  place  the 
angle  between  them  were  constant,  the  determination  of  the 
true  bearing  of  a  line  by  the  compass  would  be  comparatively 
simple.     The  needle,  however,  points  to  the  true  north  at  very 
few  places  on  the  earth's  surface,  and  the  angle  that  the  mag- 
netic meridional  plane  makes  at  any  point  with  the  true  me- 
ridional plane  is  called  the  magnetic  declination. 

75.  Variations  of  declination.     The    declination  is  subject 
at  every  place  to  changes,   called  variations   of   the   declina- 
tion.     These   are :    secular  variation,  annual  variation,  lunar 
inequalities,  and  diurnal  variation.      There  are  also  irregular 
variations  due  to  magnetic  storms.     The  annual  variation  and 
the  lunar  variation  are  both  very  small  and  may  be  neglected 
entirely.     The  diurnal  variation  is  nothing  at  about  10.30  A.M. 
and  8.00  P.M.,  and  varies  between  these  hours,  the  north  end 
being  furthest  east  at  about  8  o'clock  in  the  morning  and  fur- 
thest west  at  about  1.30  in  the  afternoon.     The  total  move- 
ment is  from  five  to  ten  minutes,  and  the  corrections  to  be 
applied  are  as  in  the  Appendix,  Table  V.,  page  364.     It  will 
be  noticed  that  the  variation  changes  with  the  seasons.     It  also 
varies  slightly  with  the  latitude,  and  the  corrections  in  the  table 
are  about  a  mean  for  the  United  States  for  the  latitudes  given. 

The  secular  variation  is  by  far  the  most  important.  It  is 
supposed  to  be  periodic  in  its  character,  requiring  from  two 
hundred  to  four  hundred  years  to  make  a  complete  cycle.1 

The  needle  in  Paris  in  1580  pointed  about  9°  or  10°  east  of 
north,  while  in  1800  it  pointed  a  little  more  than  22°  west  of 
north.  The  change  seems  to  have  been  fairly  but  not  abso- 
lutely uniform  in  rate.  The  rate  of  change  seems  to  differ 
with  difference  of  place,  and  also  with  time  at  the  same  place. 
It  is  almost  nothing  for  a  few  years  when  the  needle  is  near  its 
extreme  position  in  either  direction ;  and  again,  while  at  one 

1  For  an  extended  article  on  this  subject  see  "  United  States  Coast  and  Geodetic 
Survey  Report "  for  1882. 


MAGNETIC   DECLINATION  87 

place  the  declination  may  vary  from  one  side  of  the  meridian 
to  the  other,  at  another  place  the  entire  movement  is  on  one 
side. 

76.  Determination  of  the  declination.  Since  it  is  the  mag- 
netic meridian  that  is  given  by  the  needle,  and  since  this  is  not 
a  fixed  direction,  it  should  always  be  required  to  reduce  the 
magnetic  bearings  of  lines  run  with  the  compass  to  the  true 
bearings.  This  was  not  done  in  the  old  land  surveys  made  in 
this  country,  and  the  declination  was  not  noted.  The  result  is 
that  it  is  difficult  to  rerun  the  old  lines,  as  will  hereafter 
appear.  From  what  has  been  said  about  secular  variation,  it 
will  readily  be  seen  that  the  declination  must  be  known  for  the 
place  of  the  survey  for  the  time  of  the  survey.  In  this  country 
it  is  sufficient  to  determine  the  declination  each  year  for  a 
given  place.  It  is  done  by  the  surveyor  in  the  manner  to  be 
hereafter  described.  From  observations  conducted  over  a  num- 
ber of  years  in  many  of  the  cities  of  this  country,  Mr.  Charles 
Schott,  of  the  Coast  Survey,  has  deduced  empirical  formulas 
for  different  places  for  determining  the  declination  at  those 
places,  which  formulas  will  give  approximately  correct  results 
for  a  number  of  years  to  come.  These  formulas  are  given  in 
the  Appendix,  Table  VIII.,  pages  368-370.  The  use  of  such 
formulas  is  not  advisable  except  to  check,  in  a  general  way, 
the  results  of  the  observations  of  the  surveyor,  who  should 
always,  in  going  to  a  new  place,  determine  the  decimation  for 
himself. 

A  line  connecting  points  having  the  same  magnetic  declina- 
tion is  called  an  isogonic  line,  and  the  line  joining  points  of 
no  declination  is  an  agonic  line.  In  Plate  VI.,  at  the  end  of 
the  book,  taken  from  the  report  of  the  Coast  Survey  for  1888 
and  1889,  an  agonic  line  is  seen  extending  from  near  Charles- 
ton, S.C.,  to  the  upper  end  of  Lake  Huron,  and  passing  through 
or  near  Huntington,  W.Va.,  Newark  and  Toledo,  O.,  and  Ann 
Arbor,  Mich.  All  points  to  the  east  of  this  line  have  west 
declination,  and  all  points  to  the  west  have  east  declination. 
Moreover,  this  line  is  moving  westward,  and  hence  west  decli- 
nations are  increasing  and  east  declinations  diminishing.  The 
declination  for  a  given  place  may  be  determined  approximately 


MAGNETIC   DECLINATION.  89 

from  this  chart,  though,  from  the  extreme  irregularity  of  the 
lines  in  those  portions  of  the  country  where  observations  have 
been  most  frequent,  as,  for  instance,  in  Missouri,  it  will  be  seen 
that  too  much  reliance  can  not  be  placed  on  the  correctness  of 
the  lines.  Figure  43  is  a  chart  on  a  very  small  scale,  showing 
in  a  general  way  the  declination  at  all  points  on  the  earth. 

The  declination  is  determined  by  the  surveyor  by  first 
determining  a  true  meridian  and  then  setting  the  compass 
over  the  southern  end  of  a  line  in  this  meridian  and  reading 
the  bearing  of  the  line.  This  is,  in  amount,  the  declination. 
The  direction  is,  however,  reversed,  since  there  is  read  the 
bearing  of  the  true  meridian  referred  to  the  magnetic  meridian, 
instead  of  that  of  the  magnetic  meridian  referred  to  the  true 
meridian.  The  zeros  of  the  declination  vernier  and  its  arc 
should  coincide,  or  the  reading  will  be  erroneous.  To  read 
the  declination  more  precisely  than  can  be  done  on  the  circle, 
which  is  graduated  only  to  half  degrees,  bring  the  plane  of 
the  sights  carefully  into  the  meridian,  and  then  move  the 
vernier  by  the  tangent  screw  till  the  needle  reads  zero,  or 
north.  The  reading  of  the  vernier  will  be  the  declination. 
If  now  the  vernier  is  left  set  at  this  reading,  a  survey  may 
be  conducted  by  the  compass  and  referred  to  the  true  meri- 
dian, without  mental  reduction,  as  the  compass  will  give  the 
true  bearing  of  the  lines  at  once. 

77.  Determination  of  the  true  meridian.  The  only  real  diffi- 
culty in  finding  the  declination  lies  in  determining  the  true 
meridian.  This  is  done  in  several  ways,  one  of  which,  very  old 
and  only  roughly  approximate,  is  to  mark  on  the  ground  or  on 
a  flat  surface,  the  extremities  of  the  shadows  cast  by  a  vertical 
rod  or  other  object  when  the  sun  is  at  equal  altitudes  above 
the  eastern  and  western  horizons.  This  is  done  by  noting  the 
shadow  at  about  9  or  10  o'clock  in  the  morning  and  draw- 
ing a  circle  through  the  extremity  of  the  shadow,  taking  the 
foot  of  the  rod  as  a  center,  and  then  marking  the  point  on  this 
circle  that  is  just  touched  by  the  shadow  in  the  afternoon. 
The  point  midway  between  the  two  positions  of  the  extremity 
of  the  shadow  and  the  center  of  the  circle  lies  in  the  meridian. 

The  better  way  to  determine  the  nieridian  is  by  an  observa- 


90  DIRECTION   AND   ANGLES. 

tion  on  the  star  Polaris,  or,  as  it  is  usually  called,  the  North 
Star.  This  star  is  not  directly  at  the  north  pole,  but  is  at 
present  about  one  and  one  quarter  degrees  from  the  pole,  about 
which  it  seems  to  revolve  in  23  hours  and  56  minutes,  thus 
bringing  it  to  the  meridian  twice  every  day.  It  is  approxi- 
mately on  the  meridian  when  a  plumb  line  traverses  the  pole 
star  and  the  star  next  the  end  of  the  handle  of  the  dipper,  or 
Zeta  Ursse  Majoris,  that  is,  the  star  Zeta  in  the  constellation  of 
the  Great  Bear,  which  is  the  name  of  the  constellation  includ- 
ing the  dipper.  The  stars  in  every  constellation  are  lettered 
or  numbered.  Polaris  is  known  as  Alpha  Ursoe  Minoris,  or 
Alpha  of  the  constellation  of  the  Lesser  Bear. 

Two  thirds  of  a  minute,  1895,  after  both  of  these  stars  are 
covered  by  the  plumb  line,  Polaris  is  on  the  meridian.  For 
each  year  after  1895  add  0.35  minutes.  It  is  not  best  to  ob- 
serve the  star  at  culmination,  that  is,  on  the  meridian,  because 
at  that  time  its  motion  is  all  horizontal,  giving  a  considerable 
error  for  a  small  error  in  time  of  observation.  It  is  better  to 
observe  the  star  at  the  time  when  it  is  furthest  east  or  west, 
or,  as  is  said,  when  it  is  at  eastern  or  western  elongation.  The 
method  of  observation  is  essentially  the  same  in  both  cases. 
Suspend  a  plumb  line  from  a  beam  or  overhanging  limb  which 
is  high  enough  so  that  a  line  of  sight  to  the  star,  from  a  point 
twenty  feet  or  more  to  the  south,  will  pass  below  the  point  of 
suspension.  This  will  usually  require  a  plumb  line  at  least 
twenty  feet  long,  the  length  depending  on  the  latitude  ;  the  ele- 
vation of  the  pole  above  the  horizon  being  equal  to  the  latitude 
of  the  place.  The  top  of  the  plumb  line  should  be  illuminated 
with  a  light  screened  from  the  observer.  Select  a  point  south  of 
the  plumb  line,  approximately  in  the  meridian  and  such  that  the 
star  may  be  seen  just  below  the  top  of  the  string  ;  and  drive 
two  stout  stakes  in  an  east  and  west  line,  making  sure  that 
the  meridian  will  fall  between  the  stakes.  Drive  the  stakes  till 
their  tops  are  in  a  level  line  and  then  nail  to  their  tops  a  stiff, 
smooth  board.  Take  off  one  of  the  sights  of  the  compass  and 
rest  it  on  this  board'.  It  may  be  necessary  to  set  it  in  a  block 
because  of  the  pins  on  the  bottom.  At  a  little  before  the  time 
of  culmination  or  elongation,  whichever  is  to  be  observed,  bring 
the  sight  "  into  line  "  with  the  plumb  line  and  the  star. 


MAGNETIC   DECLINATION. 


Keep  it  in  line  by  moving  the  sight  till  the  time  of  culmina- 
tion or  elongation,  and  then  mark  on  the  board  the  position 
of  the  sight.  The  work  may  now  be  left  for  the  night.  The 
next  morning  at  a  little  after  10  o'clock,  set  up  the  compass 
over  the  point  occupied  the  previous  night  by  the  sight,  and 
turn  the  sights  on  the  point  over  which  the  plumb  line  was 
suspended.  If  culmination  has  been  observed,  the  compass  is 
now  pointing  in  the  meridian,  and  the  needle  reading  will  give, 
in  amount,  the  magnetic  declination  for  that  place. 

78.  Azimuth  at  elongation.  If  elongation  has  been  ob- 
served, the  true  bearing  of  the  star  at  elongation,  called  the 
"  azimuth "  at  elongation, 
must  be  determined,  and  a 
bearing  equal  to  this  must 
be  set  off  to  get  the  true 
meridian.  The  explanation 
of  how  this  is  done  is  as 
follows :  The  angle  that  is 
required  is  the  angle  between 
the  vertical  plane  including 
the  pole  and  the  position  of 
the  observer,  and  the  vertical 
plane  through  the  latter  point 
and  the  star  at  elongation. 
This  latter  plane  is  tangent  to 
the  circle  of  apparent  revolu- 
tion of  the  star.  If,  then,  the 
observer  were  on  the  equator, 
the  required  angle  would  be 
equal  to  the  pole  distance  of 
the  star.  As  the  position  of 
the  observer  is  moved  to  the 
north,  the  plane  becomes  tan- 
gent to  the  circle  of  revolution 
at  a  point  above  a  horizontal 
line  through  the  pole.  There-  FlQ  44. 

fore,    as   the   latitude    of    the 
observer  is  increased,  the  angle  between  the  two  planes  is  also 


92  DIRECTION   AND   ANGLES. 

increased.  These  two  planes,  extended  to  the  celestial  sphere, 
will  cut  from  that  sphere  two  arcs  which,  with  the  arc  of  the 
meridian  of  the  star  from  pole  to  star,  will  form  a  spherical 
triangle.  The  arcs  are  the  arcs  zenith-pole,  zenith-star,  and 
pole-star ;  the  triangle  formed  being  zenith-pole-star,  right- 
angled  at  star.  In  Fig.  44  the  points  are  lettered  Z,  P,  and  8. 
The  plane  of  the  terrestrial  equator  extended  to  the  celestial 
sphere  cuts  from  that  sphere  the  celestial  equator,  and  the 
zenith  of  the  place  of  observation  is  north  of  that  equator  by 
an  arc  equal  to  the  latitude  of  the  place.  Therefore,  the  pole 
is  distant  from  the  zenith  by  an  arc  equal  to  the  colatitude. 
On  the  celestial  sphere  the  term  "  declination  "  is  used  instead  of 
"  latitude,"  and  hence  the  star  is  distant  from  the  pole  by  an  arc 
equal  to  the  codeclination  of  the  star.  In  the  figure  the  arcs 
are  shown.  Calling  codeclination  of  the  pole  star  the  pole- 
distance,  and  letting  <f>  represent  the  latitude,  there  results  at 
once  from  the  principles  of  trigonometry  the  relation 

sine  pole  distance 

sine  azimuth  at  elongation  = ± — : ; 

cosine  </> 

The  declination  of  Polaris  is  constantly  increasing,  and  will 
continue  to  increase  until  the  star  is  within  about  thirty  min- 
utes of  the  pole,  when  it  will  begin  to  decrease.  Table  IV. 
Appendix,  page  364,  gives  the  polar  distance  of  Polaris  for  a 
number  of  years. 

The  latitude  of  the  place  may  be  taken  from  a  good  map, 
an  error  of  a  degree  of  latitude  involving  in  the  territory  below 
50°  north  latitude  an  error  of  2±'  in  latitude  50°,  1-*-'  in  latitude 
40°,  1'  in  latitude  30°,  and  J'  in  latitude  20°. 

The  calculation  of  the  time  of  elongation  involves  a  knowl- 
edge of  astronomical  terms  that  it  is  not  thought  wise  to 
include  here,  and  hence  the  times  of  elongation  are  given  in 
Table  VI.  Appendix,  page  365,  for  the  year  1897,  with  rules  for 
determining  the  time  for  other  years  and  other  latitudes  than 
those  for  which  the  table  was  computed. 

The  method  of  determining  the  meridian  just  given  is  suffi- 
ciently precise  for  compass  work.  A  more  precise  method  will 
be  given  in  the  discussion  of  the  transit.  As  has  been  said, 
almost  all  of  the  original  surveys  made  in  the  eastern  part 


MAGNETIC  DECLINATION.  93 

of  this  country  were  made  with  the  magnetic  compass.  Many 
of  them  were  made  and  recorded  with  reference  to  the  mag- 
netic meridian,  without  recording  the  magnetic  declination  at 
the  place  at  the  time  of  the  survey ;  and  hence,  if  it  were  re- 
quired to  retrace  one  of  the  old  surveys,  all  but  one  of  whose 
corners  had  been  lost,  it  would  be  found  to  be  a  difficult  thing 
to  do.  The  method  of  procedure  will  be  discussed  further  on. 

79.  Local  attraction.  The  needle  may  be  drawn  from  its 
mean  position  for  a  given  locality  by  the  attraction  of  large  or 
small  masses  of  iron  or  other  magnetic  substances.  These  may 
be  either  minerals  hidden  in  the  ground  or  manufactured  arti- 
cles, as  agricultural  implements  in  a  barn  near  by,  railroad 
rails,  nails  in  an  adjacent  house,  etc.  The  chain  used  in  the 
survey  may  influence  the  needle  if  brought  too  near,  while  keys 
in  the  pocket  of  the  observer,  or  steel  wire  in  the  rim  of  a  stiff 
hat,  and  steel  button  molds  on  a  coat  have  been  known  to  give 
trouble. 

Local  attraction  may  be  very  troublesome,  and  must  be 
carefully  looked  out  for.  It  may  be  discovered  and  allowed 
for  by  simply  occupying  every  corner  of  a  survey,  and  reading 
the  bearing  of  each  line  back  as  well  as  forward.  It  will  be 
evident  that  at  any  one  station  where  the  compass  is  set,  the  local 
attraction,  if  any  exists,  will  affect  by  the  same  amount  all 
readings  taken  from  that  point ;  and  hence,  if  the  bearings  of 
two  lines  meeting  at  a  point  are  both  determined  at  that  point, 
the  angle  of  intersection  is  correctly  deduced,  even  though 
there  is  local  attraction  and  neither  bearing  is  correctly  deter- 
mined. If  the  bearing  of  a  line  is  determined  from  each  end, 
and  it  is  found  that  the  two  readings  are  not  numerically  the 
same,  and  it  is  certain  that  the  bearings  are  correctly  read, 
there  is  probably  local  attraction  at  one  end  or  the  other,  or 
possibly  at  both  ends. 

If  now  another  line  is  set  off  from  one  end  of  the  first  line, 
and  the  bearing  of  the  new  line  is  read  at  both  ends,  and  the 
readings  are  found  to  agree,  the  local  attraction  affected  the 
reading  of  the  first  line  at  the  other  end.  If  trouble  is  en- 
countered on  the  second  line,  a  third  line  may  be  set  off  from 
the  other  end  of  the  first  line,  the  same  observations  being 


94  DIRECTION   AND   ANGLES. 

made  as  on  the  second  line.  If  the  auxiliary  lines  are 
properly  chosen,  one  will  usually  be  sufficient.  If  it  is  feared 
that  local  attraction  exists,  the  best  method  is  to  read  the 
bearings  of  all  lines  at  each  elid,  determining  the  angles  be- 
tween adjacent  lines  from  the  bearings  taken  at  their  points  of 
intersection.  One  line  may  then  be  assumed  to  be  correctly 
read,  or  the  true  bearing  may  be  determined,  and  the  bearings 
of  the  others  may  be  computed  from  the  series  of  angles. 

80.  Special  forms  of  compasses.  Some  compasses  are  made 
with  an  additional  full  circle  of  360°  and  a  vernier  reading 
to  minutes,  with  suitable  clamps  and  tangent  screws ;  but  it 


FIG.  45. 

may  be  said  that  the  precision  obtained  in  pointing  with 
the  open  sights  is  hardly  sufficient  to  warrant  this.  If  angles 
are  required  to  be  read  to  single  minutes,  it  is  better  to  use  a 
transit. 

The  prismatic  compass  is  a  very  convenient  instrument 
to  use  on  exploratory  work.  It  has  folding  sights,  and,  by 
means  of  the  prism,  may  be  read  while  being  pointed,  which 
is  a  convenience  when  the  instrument  is  held  in  the  hand  in- 
stead of  being  mounted  on  a  Jacob  staff  or  tripod.  It  is  usually 
held  in  the  hand,  though  it  may  be  used  either  way.  The  in- 
strument is  shown  in  Fkf.  45. 


THE  TRANSIT.  95 


THE   TRANSIT. 

81.  Description.  The  instrument  most  used  by  surveyors 
and  engineers  for  measuring  horizontal  angles,  and,  with  cer- 
tain attachments,  for  measuring  vertical  angles  and  distance, 
and  for  leveling,  is  called  a  transit. 

The  transit  consists  of  a  telescope  attached  to  a  pointer 
which  may  be  moved  around  a  graduated  circle.  There  are  suit- 
able attachments  for  controlling  the  motion  of  the  telescope 
and  the  pointer,  and  for  enabling  the  graduated  circle  to  be 
made  horizontal.  The  pointer  may  be  clamped  to  the  gradu- 
ated circle,  and  they  may  be  revolved  together.  If  this  is 
done,  and  the  telescope  is  pointed  to  a  given  object,  and  the 
graduated  circle  is  clamped  so  that  it  will  not  revolve,  and  if 
the  pofhter  is  then  undamped  from  the  circle,  the  pointer  and 
telescope  may  be  turned  together  till  the  telescope  points  to  a 
second  object.  The  number  of  degrees  of  the  circle  over  which 
the  pointer  has  passed  will  be  the  angle  subtended,  at  the  point 
where  the  transit  is  placed,  by  the  two  objects  seen  through 
the  telescope. 

The  pointer  is  the  zero  mark  of  a  vernier.  There  are  usu- 
ally two  double  verniers  placed  180°  apart. 

In  the  transit  shown  in  Fig.  46,  T  is  the  telescope  which 
may,  by  means  of  the  horizontal  axis  A,  be  revolved  in  a 
vertical  plane. 

The  horizontal  axis  rests  in  bearings  at  the  top  of  the  stand- 
ards X,  which  are  rigidly  attached  to  the  circular  plate  that 
carries  the  vernier  V. 

On  this  plate  are  set  two  level  tubes  which,  when  adjusted 
to  be  parallel  to  the  plate,  will  show  whether  the  plate  is  level. 
This  plate  is  made  by  the  maker  perpendicular  to  the  axis  on 
which  it  revolves,  and  hence,  when  the  plate  is  level,  the  axis, 
of  revolution  is  vertical.  This  axis  is  conical,  and  fits  inside  a 
conical  socket  which  is  the  inside  of  the  conical  axis  of  the  plate 
that  carries  the  graduated  circle.  This  latter  conical  axis  re- 
volves in  a  socket  that  connects  the  top  and  bottom  plates  of 
the  leveling  head.  The  upper  plate  is  leveled  by  the  leveling 
screws  L.  The  lower  plate  of  this  leveling  head  screws  on  to 
the  tripod.  The  clamp  C  fastens  together  the  vernier  plate 


96 


DIRECTION   AND   ANGLES. 


(sometimes  called  the  alidade^)  and  the  plate  carrying  the 
graduated  circle.  The  latter  plate  is  called  the  limb.  When 
the  two  plates  are  clamped  together,  the  vernier  plate  may  still 


FIQ.  46. 


be  moved  a  small  amount,  relatively  to  the  limb,  by  means  of 
the  tangent  screw  S.  This  is  for  the  purpose  of  setting  the 
vernier  at  a  given  reading,  or  pointing  the  telescope  at  a  point, 


THE   TRANSIT.  97 

more  precisely  than  would  be  ordinarily  possible  by  simply 
turning  the  alidade  by  the  hand.  The  collar  K  surrounds  the 
spindle  of  the  limb,  and  by  the  clamp  screw  C'  may  be  fastened 
to  that  spindle.  The  lug  M,  which  is  attached  to  the  collar, 
being  held  by  the  spring  in  the  barrel  P  and  the  opposing 
screw  /S",  in  turn  fastened  to  the  upper  leveling  plate,  prevents 
the  revolution  of  the  limb,  when  the  collar  is  clamped  to  its 
axis.  The  limb  may,  however,  be  moved  a  small  amount  by 
the  tangent  screw  S'  working  against  the  pressure  of  the 
spring  in  the  barrel  P. 

Some  instruments  when  put  away  in  the  box  in  which  they 
are  kept  when  not  in  use,  are  separated  into  two  parts.  The 
upper  part  of  the  instrument  is  separated  from  the  leveling 
head,  and  the  two  parts  are  placed  separately  in  the  box. 
Others  are  so  made  as  to  be  put  away  in  one  piece,  being 
merely  unscrewed  from  the  tripod  and  screwed  to  a  board 
that  slides  into  the  box. 

There  are  various  forms  of  transits  and  various  patterns 
for  the  different  parts,  according  to  the  ideas  of  the  different 
makers.  A  careful  examination  of  an  instrument  should  be 
made  before  putting  it  to  use,  that  the  user  may  become  per- 
fectly familiar  with  its  construction. 

The  circles  are  usually  so  graduated  that  angles  may  be  read 
to  minutes.  Many  instruments  are  graduated  to  read  30  sec- 
onds, some  to  read  20  seconds,  and  a  few  to  read  10  seconds. 
For  ordinary  land  surveying,  single  minutes  are  sufficiently 
precise,  while  for  fine  city  surveying  20  seconds,  or  even  10 
seconds,  is  not  too  fine. 

The  telescope  is  essentially  the  same  as  that  used  in  the 
level,  but  is  shorter  and  usually  of  lower  magnifying  power. 
It  should  be  inverting,  but  is  usually  an  erecting  glass.  The 
use  of  the  inverting  glass  is  growing.  There  is  a  proper  rela- 
tion between  the  magnifying  power  of  the  telescope  and  the 
least  count  of  the  vernier  used  ;  and  it  is  not  necessarily  the 
best  instrument  that  has  the  smallest  count  vernier,  for  the  rea- 
son that  the  magnifying  power  of  the  telescope  may  be  such 
that  a  movement  of  several  of  the  smallest  units  that  can  be 
read  by  the  vernier  may  give  no  perceptible  lateral  motion  to 
the  line  of  collimation.  The  power  of  the  telescope  and  the 
II'M'D  suuv.  — 7 


98 


DIRECTION   AND   ANGLES. 


least  count  of  the  vernier  should  be  so  adjusted  to  each  other  that 
a  barely  perceptible  movement  of  the  vernier  will  cause  a  barely 
perceptible  movement  of  the  line  of  collimation.  The  same  is 
true  of  the  magnifying  power  and  the  level  under  the  telescope. 
Methods  of  testing  these  relations  will  suggest  themselves. 

The  verniers  are  usually  read  with  small  magnifying  glasses 
known  as  reading  glasses. 

In  order  that  the  transit  may  be  set  precisely  over  a  point 
on  the  ground,  there  is  fastened  to  the  center  of  the  lower  part 
of  the  leveling  head  a  ring  or  hook,  from  which  may  be  sus- 
pended a  plumb  line.  A  sectional  view  of  the  lower  part  of  the 
transit  in  Fig.  46  is  shown  in  Fig.  47.  It  will  be  seen  that  the 


FIG.  47. 

upper  leveling  plate  U — in  some  forms  four  arms  instead  of 
a  plate — is  attached  to  the  socket  in  which  the  vertical  axis  re- 
volves, and  that  to  this  upper  plate  are  attached  the  nuts  in 
which  the  leveling  screws  work.  These  leveling  screws  rest  in 
little  cups,  that  in  turn  rest  on  the  lower  leveling  plate  L. 
The  ball  and  socket  jpint  that  permits  the  leveling  of  the  instru- 
ment is  separate  from  the  lower  leveling  plate,  and  is  not  quite 
so  large  as  the  hole  in  the  center  of  that  plate.  There  is  an 
extension  plate  to  this  ball  and  socket  joint  that  extends  under 
the  lower  leveling  plate.  When  the  leveling  screws  are  tight- 
ened, this  extension  plate  binds  against  the  leveling  plate,  and 


THE   TRANSIT.  99 

the  instrument  can  not  be  shifted  on  the  lower  plate.  If  the 
leveling  screws  are  loosened,  the  upper  part  of  the  instrument 
may  be  shifted  laterally  on  the  lower  plate  by  an  amount  de- 
pending on  the  relative  diameters  of  the  ball  joint  and  the  hole 
in  the  center  of  the  lower  plate.  The  device  is  called  a  shifting 
center  and  is  found  in  all  modern  transits. 

82.  The  tachymeter.  Such  a  transit  as  has  been  described, 
while  still  in  very  general  use,  is  being  rapidly  displaced  by 
the  complete  engineer's  transit,  or  tachymeter.  This  instru- 
ment consists  of  such  a  transit  as  has  been  described,  with  the 
following  attachments : 

(1)  A  level  under  the  telescope,  making  the  transit  a  good 
leveling  instrument. 

(2)  A  vertical  circle  or  arc,  by  which  vertical  angles  may  be 
read. 

(3)  A  gradienter  attachment  by  which  very  small  vertical 
angles  may  be  set  off,  as  so  many  feet  rise  or  fall  in  one  hundred. 

(4)  A  pair  of  horizontal  wires  in  the  telescope,  known  as 
stadia  wires. 

There  are  various  other  attachments  for  special  purposes 
that  will  be  found  described  in  instrument  makers'  catalogues, 
but  which  it  is  not  necessary  to  mention  here.  There  are  also 
special  forms  of  transits  for  use  underground.  One  form  of 
the  complete  instrument  is  shown  in  Fig.  48. 

The  level  under  the  telescope  and  the  vertical  circle  are 
readily  recognized.  The  gradienter  and  slow-motion  screw,  for 
vertical  motion  of  telescope,  is  shown  at  G.  The  telescope  is 
clamped  in  vertical  motion  by  the  clamp  C,  It  may  then  be 
moved  slowly  a  small  amount  by  the  slow-motion  screw.  The 
head  of  this  screw  is  graduated  with  reference  to  the  pitch  of 
the  screw,  so  that  a  single  revolution  of  the  head  corresponds  to 
an  angular  motion  of  the  telescope  of  one  foot  in  one  hundred,  or 
half  a  foot  in  one  hundred.  In  the  figure,  the  head  is  graduated 
in  the  latter  way,  as  is  indicated  by  the  double  row  of  figures. 
The  use  of  the  gradienter  is  chiefly  in  setting  out  grades. 

Many  transits  are  fitted  with  the  slow-motion  screw  without 
the  gradienter  head.  All  makers  make  both  the  plain  and  com- 
plete transits. 


100 


DIRECTION    AND    ANIJI.KS. 


USE  OF  THE  TRANSIT. 


83.  Carrying  the  transit.  While  in  use,  the  transit  is  not 
removed  from  the  tripod  when  it  is  carried  from  one  point  to 
another;  but  is  carried  with  the  tripod  on  the  shoulder  of  the 


FIG.  48. 

"  transitman."     The  plumb  bob  is  carried  in  the  pocket  without 
removing  the  string  from  the  instrument.     Before  the  transit 


USE   OF   THE   TRANSIT.  ;()». 

can  be  properly  set  up  and  the  angles  measured,  the  various 
parts  must  be  in  proper  adjustment.  The  discussion  of  the 
adjustments  will,  however,  be  deferred  till  the  use  of  the  ad- 
justed transit  has  been  described. 

84.  "Setting  up"  the  transit.     To  "set   up"  the  transit 
over  a  point,  let  the  plummet  swing  free,  grasp  in  one  hand 
the    uppermost    leg   of    the   tripod,    and    plant    it    firmly    in 
the  ground.      Next  grasp   the   other  two  legs,  one   in'  each 
hand,  and  place  them   symmetrically  about  the  point  so  that 
the    plumb    bob    covers    the    point    as    nearly    as    may    be. 
The  plates  should  be  at  the  same  time   observed   arid   made 
approximately  level.      It   should   be   noticed   that  a  sidewise 
motion  of  a  leg  changes  the  level  of  the  plates  without  much 
disturbing  the   plumb  bob,  and  that  a  radial  motion  changes 
the  level  of  the  plates  to  a  less  degree,  but  does  disturb  the 
plumb  bob.      It  is  necessary  to  become  expert  in  setting  the 
transit,  as  much  time  may  be  lost  by  awkward  work  in  setting 
up.      After  the   transit  is  set  approximately  over  the  point, 
press  the  legs  firmly  into  the  ground  so  as  to  bring  the  bob 
more  nearly  over  the  point  and  to  make  the  instrument  firmer. 
Finish  centering  by  the  adjustable  or  shifting  center,  by  loos- 
ening all  the  leveling  screws  and  moving  the  whole  upper 
part  of  the  instrument  till  it  is  centered.     If  there  is  no  ad- 
justable center,  the  instrument  must  be  centered  by  manipu- 
lating the  legs.     When  the  instrument  is  finally  centered,  level 
the  limb  by  the  leveling  screws.     To  do  this,  place  one  plate 
bubble  parallel  to  one  set  of  opposite  leveling  screws;    then 
the  other  bubble  will  be  parallel  to  the  other  set.     Level  one 
bubble  at  a  time  and  note  that  leveling  one  will,  to  a  slight 
extent,  disturb  the  other,  which  must  then  be  leveled  again. 
If  the  lower  plate  is  not  fairly  level,  u  leveling  up "  will  dis- 
turb the  centering  of  the  plumb  bob,  which  must  then  be  cor- 
rected.    Continue  till  both  bubbles  are  in  the  centers  of  the 
tubes.     Set  the  vernier  that  is  to  be  used  at  the  proper  reading 
(this  will  often  be  zero),  and  the  instrument  is  set  up. 

85.  To  produce  a  straight  line  with  the  transit.     Set   the 
instrument  over  one  extremity  of  the  line,  and  with  the  limb 
free  to  turn,  turn  the  telescope  on  the  other  extremity  or  on 


102  DIRECTION  AND  ANGLES. 

a  point  in  the  line.  Clamp  all  motions  except  the  vertical 
motion  of  the  telescope,  and,  with  either  of  the  horizontal  tan- 
gent screws,  but  preferably  with  the  lower  screw,  bring  the  line 
of  collimation  to  bisect  exactly  the  distant  point.  Transit  the 
telescope  and  set  a  point  beyond  it  at  any  desired  place,  in  the 
line  of  collimation.  This  point  will,  if  the  instrument  is  in 
adjustment,  be  in  the  straight  line  produced.  If  it  is  feared 
that  the  transit  is  slightly  out  of  adjustment,  unclamp  the  limb, 
turn  the  instrument  till  the  line  of  collimation  again  cuts  the  first 
point,  clamp  and  set  precisely,  and  then  transit  and  set  a  new 
point  beside  the  first  one  set ;  the  point  midway  between  the  two 
set  will  be  a  point  in  the  continuation  of  the  straight  line.  This 
is  called  "  double  centering,"  and  is  the  same  operation  as  the  test 
of  the  adjustment  of  the  vertical  wire,  to  be  hereafter  explained. 
The  method  of  ranging  out  a  number  of  points  in  a  straight  line 
in  the  same  direction  will  be  apparent  without  explanation. 

86.  To  measure  an  angle  with  the  transit.     Set  the  instru- 
ment over  the  vertex  of  the  angle,  with  the  vernier  at  zero. 
With  the  lower  motion  bring  the  line  of  collimation  to  bear 
approximately  on  one  of  the  distant  points,  clamp,  and  with 
the  lower  tangent  screw  make  an   exact   bisection.      Loosen 
the  alidade  and  bring  the  line  of  collimation  approximately 
to  the  second  point,  clamp,  and  complete  the  bisection  with 
the  upper  tangent  screw.      Read  the  vernier.      The  reading 
will  be  the  angle   sought.     This  assumes  the   numbering   to 
extend  both  ways  from  0°  to  360°.     It  is,  of  course,  unneces- 
sary to  set  the  vernier  at  zero.      When  the  instrument  has 
been   set   on   the   first   point,   a   reading   may   be   taken   and 
noted  and  subtracted  from  the  reading  found  on  pointing  the 
telescope  to  the  second  point.     In  this  case  the  vernier  may 
pass  the  360°  point,  and  it  will  then  be  necessary  to  add  360° 
to  the  final  reading  before  subtracting  the  first. 

87.  Azimuth.     The  azimuth  of  a  line  is  the  horizontal  angle 
the  line  makes  with  a  reference  line,  as  a  meridian.     It  differs 
from  bearing,  in  that  it  is  measured  continuously  from  0°  to 
360°.     If  the  azimuth  of  a  point  is  mentioned,  there  is  implied 
another  point  and  a  meridian  through  it  from  which  the  azi- 
muth is  measured. 


USE   OF   THE   TRANSIT. 


103 


In  making  surveys  with  the  transit,  except  possibly  city 
surveys  and  others  where  angles  are  to  be  repeated  several 
times,  it  is  better  to  use  azimuths  than  bearings.  It  is  neces- 
sary, however,  in  writing  descriptions  of  property,  to  reduce 
the  azimuths  to  bearings. 

It  is  customary  to  reckon  azimuths  from  the  south  point 
around  by  way  of  the  west,  north,  east,  south,  360°.  This  is 
the  practice  of  astronomers  and  geodesists.  It  is  believed  to 
be  more  convenient  for  the  surveyor  to  begin  with  zero  at  the 
north  point  and  read  to  the  right  360°.  The  reason  for  this 
will  appear  in  Chapter  VI. 

88.  Traversing.  The  method  of  traversing  with  the  transit 
is  as  follows  :  In  Fig.  49,  let  it  be  required  to  determine  the 


FIG.  49. 

lengths  and  azimuths  of  the  courses  of  the  crooked  road  A,  B, 
(7,  Z>,  etc.  It  will  be  assumed  that  the  magnetic  meridian  SN 
is  the  meridian  of  the  survey.  Set  the  transit  over  a  tack  in  a 
stake  driven  at  A,  and,  with  the  vernier  set  at  zero,  turn  the 
telescope  by  the  lower  motion  in  the  direction  AN  as  denned 
by  the  needle,  and  clamp  the  lower  motion.  For  the  purpose 
of  again  using  this  line,  if  necessary,  set  a  stake  in  the  line 
AN,  some  distance  toward  N,  and  put  a  tack  in  it  in  the  exact 
line.  This  is  necessary  because  the  needle  would  not  give  the 
direction  twice  alike  to  the  nearest  minute,  which  is  desired  in 
transit  work.  Unclamp  the  alidade  and  turn  the  telescope 
toward  a  point  in  the  first  turn  of  the  road,  as  B.  Clamp  the 
alidade  and  set  a  stake  at  B.  In  most  transit  work  all  stakes 
marking  stations  to  be  occupied  by  the  transit  are  "  centered  " 
by  setting  a  tack  in  the  precise  line.  Read  and  record  the 
vernier  and  the  needle.  They  should  agree  within  the  limit  of 
precision  of  the  needle  reading.  The  angle  read  is  that  which 
has  been  turned  to  the  right,  and  in  the  figure  is  greater  than 
270°.  (The  mistake  of  reading  the  wrong  circle  must  be 


104  DIRECTION   AND   ANGLES. 

avoided.  The  needle  reading  may  be  recorded  as  bearing  or 
may  be  mentally  reduced  to  azimuth  and  so  recorded.  If  the 
compass  box  is  graduated  continuously  360°,  the  needle  will 
give  azimuths  at  once.) 

Measure  AB,  and,  while  this  is  being  done,  take  the 
transit  to  B  and  set  up  over  the  tack  that  has  been  put 
in  the  stake.  With  the  same  vernier  used  at  A  (let  it 
be  called  vernier  A)  bring  the  instrument  to  read  the  azi- 
muth AB  plus  180°,  which  is  the  "back  azimuth"  of  AB. 
Point  the  telescope  to  A  by  using  the  lower  motion,  clamp, 
and  make  the  exact  pointing  with  the  lower  tangent  screw. 
By  so  doing  the  instrument  is  oriented;  that  is,  the  zeros 
of  the  limb  are  made  parallel  to  their  positions  at  A.  (The 
telescope  has  not  been  transited,  and,  if  the  alidade  is  now 
undamped  and  the  telescope  turned  in  azimuth  just  180°, 
the  line  of  sight  will  be  in  AB  prolonged,  the  vernier  will 
read  the  azimuth  of  AB,  and  the  zeros  of  the  limb  will  be 
seen  to  lie  in  the  meridian  of  the  survey.  This  intermediate 
step  is  not  usually  taken  separately  in  practice,  but  is  intro- 
duced here  for  the  clearer  understanding  of  the  method.) 

With  the  alidade  undamped,  turn  the  telescope  in  the 
direction  of  the  next  point,  as  (7.  Clamp  the  alidade,  set  a 
stake  and  tack  at  (7,  and  read  and  record  the  vernier  and  needle. 
The  reading  of  the  vernier  will  be  the  azimuth  of  BC  referred 
to  AN,  and  the  needle  should  read  the  same  or  the  correspond- 
ing bearing.  Measure  BC,  take  the  transit  to  (7,  set  the  vernier 
at  the  back  azimuth  of  BO,  and  proceed  as  at  B.  Thus  con- 
tinue the  work  till  the  traverse  is  complete. 

89.  Transit  vs.  compass.  The  form  for  the  field  notes  of 
the  survey  just  outlined  is  the  same  as  for  a  compass  traverse, 
except  that  azimuths  instead  of  bearings  are  recorded  and 
another  column  is  used  in  which  to  record  the  needle  readings 
as  a  future  evidence  of  the  correctness  of  the  work.  Any  land 
survey  may  be  made  with  the  transit,  the  object  being,  of 
course,  the  same  as  if  made  with  the  compass.  The  essential 
difference  is  that  the  work  is  done  with  a  greater  degree  of 
precision  with  the  transit  and  azimuths  are  used  instead  of 
bearings.  One  difference  that  should  also  be  noted  is  this : 


USE  OF   THE   TRANSIT.  105 

the  use  of  the  compass  makes  every  line  independent  of  every 
other  line,  so  far  as  direction  is  concerned,  since  the  direction 
of  each  line  is  determined  independently  by  the  needle.  With 
the  transit  this  is  not  so.  An  error  in  one  line  is  carried  through 
the  remainder  of  the  survey ;  and,  being  an  angular  error,  the 
error  of  position  of  the  final  point  of  the  survey  is  much  greater 
than  it  would  be  if  there  were  an  error  in  the  direction  of  bat 
one  line  of  the  survey.  It  may  be  said,  however,  that  with  the 
needle  check  always  applied,  there  is  very  little  probability  of 
an  error  of  this  kind,  and  the  use  of  the  transit  is  advised  in  all 
work  of  importance.  Where  speed  and  roughly  approximate 
results  are  the  chief  requisites,  the  compass  is  the  better  instru- 
ment to  use,  unless  the  measurements  may  be  made  with  the 
stadia,  in  which  case  the  transit  still  remains  the  better  tool.  The 
two  may  be  combined.  The  directions  may  be  determined  with 
the  compass  attached  to  the  transit,  and  the  distances  read  with 
the  stadia,  thus  securing  a  maximum  economy  of  time.  In  this 
case  the  stakes  would  not  be  centered  with  tacks,  one  or  two 
inches  making  little  error  in  such  work,  since  the  compass  may  not 
be  read  to  less  than  five  minutes,  and  this  by  estimation.  Five 
minutes  of  arc  means  0.15  of  a  foot  in  one  hundred  feet.  More- 
over, the  distances,  if  read  by  the  stadia,  may  not  be  determined 
closer  than  the  nearest  half  foot  or  possibly  the  nearest  foot. 

90.  Determining  the  meridian.  In  many  surveys  it  is  not 
necessary  or  common  to  determine  the  meridian  before  the 
survey  is  made.  One  line  of  the  survey,  usually  the  first  run, 
is  assumed  as  a  meridian,  and  the  direction  in  which  the  survey 
proceeds  along  this  line  is  assumed  as  zero  azimuth.  The 
azimuths  of  all  lines  of  the  survey  are  then  determined  with 
reference  to  this  one  line.  If,  for  any  reason,  it  is  desired  to 
know  the  true  bearing  or  azimuth  of  the  lines  of  the  field  or 
survey,  an  observation  for  the  meridian  is  made,  and,  when  the 
meridian  is  determined,  the  angle  that  it  makes  with  one  side 
of  the  survey  is  found  with  the  transit,  and  all  the  azimuths 
are  corrected  by  this  angle  to  make  them  read  from  the  true 
meridian.  The  method  of  determining  the  meridian  with  the 
transit  is  the  same  as  with  the  compass,  except  for  the  differ- 
ence in  pointing. 


106  DIRECTION  AND   ANGLES. 

The  time  of  the  elongation  of  Polaris  is  determined,  and  a 
few  minutes  before  that  time  the  transit  is  set  up  over  a  stake 
in  an  open  space  where  the  star  may  be  observed.  As  this 
observation  is  usually  made  at  night,  it  is  necessary  to  make 
some  provision  for  illuminating  the  wires  in  the  transit  so  that 
they  may  be  seen.  This  is  usually  done  by  throwing  a  faint 
light  into  the  object  end  of  the  telescope  by  holding  a  piece 
of  paper  a  little  to  one  side  and  before  it,  and  a  lamp  —  a  bull's 
eye  is  best — behind  it  so  as  to  get  a  reflection  of  light  from 
the  paper  in  the  telescope.  It  is  troublesome  to  do  this,  and  it 
takes  two  or  three  observers.  A  better  way  is  to  have  a  re- 
flector like  one  of  those  in  Fig.  50,  which  fits  on  to  the  object 
end  of  the  telescope  and  is  illuminated  by  a  lamp 
held  back  of  the  observer. 
A  still  better  way,  when  the 
instrument  is  made  for  it, 
is  to  have  the  horizontal  axis 
FlG-  50-  of  the  transit  hollow,  with  a 

small  mirror  near  the  center  of  the  telescope.  A  small  bull's- 
eye  lamp  is  placed  on  a  stand  that  is  fastened  to  the  standards 
and  throws  a  beam  of  light  upon  the  mirror,  by  which  it  is 
reflected  to  the  wires.  The  light  must  not  be  too  strong  or 
the  star  will  be  indistinct.  This  is  particularly  true  when, 
as  may  be  the  case,  another  and  less  bright  star  than  Polaris 
is  used. 

The  transit  being  set  up  and  provision  being  made  for  illu- 
minating the  wires,  the  telescope  is  turned  on  the  star  and  both 
plates  are  clamped.  The  vertical  wire  is  then  made  to  cover  the 
star  as  nearly  as  possible  and  is  made  by  the  tangent  screw  of 
the  alidade  to  follow  the  star  as  it  seems  to  move  to  the  right 
or  left,  that  is,  in  azimuth,  until  it  seems  to  be  stationary  in 
azimuth  and  to  be  moving  only  vertically.  The  telescope  is 
then  plunged  and  a  point  set  in  the  ground  some  distance  away 
and  left  till  morning. 

The  setting  of  the  point  requires  some  patience.  Perhaps 
the  best  way  to  accomplish  this  is  to  provide  a  box  open  on  two 
sides.  Cover  one  side  with  thin  tissue  paper,  and  place  a 
candle  in  the  box.  This  improvised  lantern  may  be  approxi- 
mately set  from  the  transit  in  the  proper  line,  and  the  point  to 


USE   OF   THE   TRANSIT.  107 

drive  the  stake  may  be  determined  by  holding  a  plumb  line  in 
front  of  the  box,  the  papered  side  being  turned  toward  the 
transit.  This  line  will  be  put  in  position  from  the  instrument 
and  the  stake  driven.  The  stake  will  then  be  centered  by  the 
use  of  the  line,  and  a  tack  driven.  On  the  next  day  at  about  10 
o'clock  the  transit  is  set  over  the  point  occupied  the  previous 
evening;  and  the  azimuth  of  Polaris  at  elongation,  which  has  been 
previously  computed  as  explained  on  page  364,  is  turned  off  from 
the  line  of  stakes  to  the  right,  if  western  elongation  has  been 
observed,  and  to  the  left,  if  eastern  elongation  has  been  observed. 
Another  stake  is  now  set  in  the  line  thus  determined  and  the 
line  denned  by  the  point  occupied  by  the  transit,  and  the  last 
stake  will  be  the  true  meridian. 

With  the  instrument  set  on  this  line  the  reading  of  the 
needle  will  give  the  magnetic  declination  in  amount,  but  with 
the  opposite  sign.  That  is  to  say,  the  needle  will  read  the  mag- 
netic bearing  of  the  true  meridian  ;  and  if  the  magnetic  meridian 
lies  west  of  the  true  meridian,  thus  making  the  declination  west, 
the  needle  will  read  the  bearing  of  the  true  meridian  as  east. 

91.  Needle  checks  on  azimuths.  In  making  surveys,  if  the 
compass  box  is  not  provided  with  a  declination  plate  or  ver- 
nier, on  which  the  declination  may  be  set  off  so  that  the  needle 
will  read  north  when  pointing  in  the  true  meridian,  it  is  often 
convenient  to  make  the  meridian  of  the  survey,  the  magnetic 
meridian  of  the  place,  to  facilitate  checking  by  the  needle  the 
angles  measured  with  the  transit.  It  is  inconvenient  to  do  this 
even  then,  because  the  compass  is  graduated  from  two  points 
90°  each  way,  while  for  azimuths,  the  transit  is  graduated  con- 
tinuously through  360°.  This  difficulty  is  obviated  by  making 
a  similar  graduation  on  the  compass  box.  If  one  has  a  transit 
that  is  not  graduated  in  this  way,  he  may  graduate  a  paper 
ring  and  paste  it  on  the  glass  cover  of  the  box.  The  gradua- 
tions need  be  only  to  degrees.  It  will  be  found  that  such  a 
check  on  angle  measurements  is  very  valuable.  If  the  ring  has 
been  properly  set,  and  if  the  magnetic  meridian  has  been  chosen 
as  the  meridian  of  the  survey,  the  reading  of  the  compass 
should  always  agree  practically  with  that  of  the  transit.  If, 
instead  of  the  magnetic  meridian  some  arbitrary  meridian  has 


108  DIRECTION  AND  ANGLES. 

been  assumed  as  the  meridian  of  the  survey,  the  paper  circle 
should  be  placed  so  that  the  needle  will  read  zero  when  the 
telescope  is  pointed  in  the  direction  of  the  assumed  zero  azi- 
muth, the  vernier  of  the  transit  reading  zero.  The  needle 
check  is  good  to  the  nearest  five  minutes  except  where  there 
is  local  attraction. 

Some  transits  have  the  graduations  of  the  limb  numbered 
as  are  those  of  the  compass  box.  This  is  an  old  method.  It 
is  better  to  provide  continuous  numbering  on  the  compass  box. 

ADJUSTMENT  OF  THE  TRANSIT. 

92.  Requirements.     The  transit  is  used  for  measuring  hori- 
zontal angles,  that  is,  angles  subtended  at  a  given  point   by 
the  vertical  planes   through   the   two    other   observed   points. 
If  these  two  other  points  are  not  in  the  same  horizontal  plane, 
the  vertical  motion  of  the  telescope  must  be  used  to  bring  the 
line  of  sight  down  from  the  higher  point  to  the  horizontal 
plane  of  the  instrument  and  the  line  of  sight  of  the  lower  point 
up  to  the  same  plane.     Now,  the  line  of  collimation  is  the  line 
in  the  instrument  that  is  directed  to  the  distant  point,  and  the 
line  that  is  revolved  down  or  up,  as  the  case  may  be,  to  the 
horizontal  plane   of   the   instrument,   which  is  the  horizontal 
plane  through  the  center  of  the  horizontal  axis.     It  is  evident 
that  this  line  must  revolve  in  a  vertical  plane,  to  properly  pro- 
ject the  distant  points  into  the  horizontal  plane  of  the  instru- 
ment.    It  will  also  be  evident  that  the  vertical  axis  of  the 
transit  must  be  truly  vertical  in  order  that  the  line  of  sight 
when  projected  into  the  horizontal  plane  of  the  instrument  and 
then  turned  in  azimuth,  may  move  in  a  horizontal  plane.     If 
the  axis  of  revolution  is  not  vertical,  the  lateral  motion  of  the 
instrument  will  be  in  an  inclined  plane.     In  order  that  these 
necessary  conditions    may  obtain,   certain  adjustments  of  the 
instruments  must  be  properly  made.     Every  adjustment  con- 
sists of   two  parts :    the  test  to  determine  the  error,  and  the 
rectification  of  the  error  found. 

93.  The  plate  bubbles.     If  the  plate  bubbles  are  perpen- 
dicular to  the  axis  of  revolution,  that  axis  will  be  vertical  when 
both  bubbles  are  in  the  centers  of  their  respective  tubes.     The 


ADJUSTMENT   OF   THE   TRAXSIT.  109 

test  is  made  and  the  adjustment  is  performed  as  described  for 
the  compass.  If  one  bubble  gets  broken,  the  other  may  be  used 
for  both,  by  first  leveling  with  the  bubble  in  one  position  and 
then  in  another,  90°  from  the  first,  exactly  as  in  setting  up  the 
level,  and  by  repeating  the  operation  till  the  axis  is  vertical. 
But  two  operations  would  be  necessary  were  it  not  for  the 
fact  that  the  second  leveling  will,  to  some  extent,  disturb  the 
first. 

The  adjustment  being  made,  all  points  of  the  instrument 
will  revolve  in  horizontal  planes  if  the  bubbles  are  both  in  the 
centers  of  their  tubes.  This  adjustment  should  be  tested 
every  day,  although  it  will  probably  be  found  correct  for  a 
considerable  number  of  days  in  succession.  If  the  transit  is 
out  of  adjustment  all  around,  it  is  better  to  make  each  adjust- 
ment approximately  in  order  and  then  carefully  repeat  them 
all. 

94.  To  make  the  line  of  collimation  revolve  in  a  vertical 
plane  when  the  telescope  is  turned  on  its  horizontal  axis.  This 
adjustment  consists  of  several  parts.  One  of  these  is  to  make 
the  axis  of  revolution  of  the  telescope  horizontal.  If  the  in- 
strument is  provided  with  a  striding  level,  this  part  may  be 
performed  first.  If  not,  as  is  usually  the  case,  it  must  be  done 
last,  and  possibly  at  the  expense  of  repeating  some  of  those 
adjustments  that  have  preceded  it.  It  will  be  assumed  that  no 
striding  level  is  used. 

If  the  line  of  collimation  is  not  perpendicular  to  the  axis  of 
revolution  of  the  telescope,  it  will  describe,  in  revolving,  the 
surface  of  a  cone,  whose  axis  is  the  horizontal  axis  of  revo- 
lution of  the  telescope  ;  and  if  the  line  of  collimation  is  per- 
pendicular to  the  axis  and  the  axis  is  not  horizontal,  the  line 
of  collimation  will  describe,  on  being  revolved,  an  inclined 
instead  of  a  vertical  plane.  The  student  should  make  a  mental 
picture  of  these  conditions. 

The  intersection  of  the  wires  should  be  in  the  line  of  motion 
of  the  optical  center  of  the  objective ;  for,  if  not,  then  the  line 
of  collimation  will  not  be  fixed  in  its  position  in  the  telescope 
tube,  and,  if  made  perpendicular  to  the  axis  of  revolution  for 
one  distance,  it  would  not  be  so  for  some  other  distance  that 


110  DIRECTION  AND   ANGLES. 

would  require  a  shifting  of  the  object  slide  for  new  focusing. 
The  object  slide  should  move  parallel  to,  if  not  absolutely  coin- 
cident with,  the  axis  of  the  telescope  tube,  in  order  that  the 
line  of  collimation  may  be  made  perpendicular  to  the  horizontal 
axis  of  revolution.  Theoretically,  its  line  of  motion  should 
pass  through  the  horizontal  axis.  In  the  instrument  shown  in 
Fig.  48  the  objective  is  permanently  adjusted  in  motion  by  the 
maker.  In  some  other  instruments,  however,  this  is  not  the 
case. 

The  two  wires  are  adjusted  separately,  and  it  is  not  unusual 
to  omit  the  adjustment  of  the  horizontal  wire.  This  should 
not  be  omitted  if  the  instrument  is  to  be  used  for  leveling  or 
for  measuring  vertical  angles.  The  first  adjustment  of  the 
vertical  wire  is  to  make  it  vertical.  This  can  be  done  only 
approximately,  if  the  horizontal  axis  has  not  been  previously 
adjusted. 

I.  To  make  the  vertical  wire  vertical.     Carefully  focus  the 
eyepiece,  level  the  instrument,  turn  the  telescope  on  a  sus- 
pended  plumb   line,   and   observe    whether   the   vertical  wire 
coincides  or  is  parallel  with  it.     The  corner  of  a  plumb  build- 
ing will  sometimes  answer  the  purpose  if  the  wind  interferes 
too  much  with  the  swinging  plumb  line.     If  the  wire  is  found 
to  be  out  of  plumb,  it  should  be  made  plumb  by  loosening  all 
four  of  the  capstan-headed  screws  that  hold  the  ring  carry- 
ing the  wire,  and  by  moving  the  ring  around  by  the  screws 
till  the  wire  is  vertical.     The  holes  by  which  the  adjusting 
screws  pass  through  the  telescope  tube   are  slotted  for  this 
purpose. 

The  remaining  part  of  the  adjustments  will  be  described  as 
it  is  considered  best  to  make  them  in  the  complete  transit  or 
tachymeter.  (See  Art.  114,  page  127.)  The  adjustments  that 
are  to  be  made  when  only  the  plain  transit  is  to  be  adjusted 
are  those  of  Art.  93  and  numbers  I.,  III.,  and  IV.,  of  this 
article. 

II.  To  make  the  line  of  collimation  parallel  to  or  coincident 
with  the  geometric  axis  of  the   telescope.    Construct  of  wood  a 
pair  of  Y  standards  fastened  to  a  firm  block  of  wood  and  with 
such  a  distance  between  them  that  the  telescope  tube  may, 
after  being  removed  from  its  standards,  rest  in  the  Y's  near  its 


ADJUSTMENT   OF   THE   TRANSIT. 


Ill 


ends.  Having  made  the  vertical  wire  vertical  as  described  in 
the  last  adjustment,  remove  the  telescope  with  its  horizontal 
axis  from  the  standards,  and  rest  it  .in  the  Y  supports.  Now 
adjust  the  wires  (and  the  object  slide  if  it  is  adjustable 
and  needs  it)  in  precisely  the  same  manner  as  described  in  the 
first  adjustment  of  the  level.  If  the  geometric  axis  of  the  tele- 
scope has  been  made  exactly  perpendicu- 
lar to  the  horizontal  axis,  and  the  two  are 
coincident  where  they  cross,  the  line  of 
collimation  is  now  perpendicular  to  the 
horizontal  axis.  The  foregoing  adjust- 
ment is,  in  all  good  instruments,  sufficient 
for  the  horizontal  wire.  After  replacing 
the  telescope  in  its  standards,  the  vertical 
wire  is  tested  and,  if  necessary,  corrected 
as  explained  in  III. 

III.  If  the  line  of  collimation  as  de- 
fined by  the  optical  center  of  the  objective 
and  the  vertical  wire  (which  is  the  wire 
most  used  in  the  transit)  is  not  perpen- 
dicular to  the  horizontal  axis  of  revolu- 
tion, and  the  instrument  is  set  over  the 
middle  one  of  three  points  that  are  in  a 
straight  line,  the  conditions  will  be  as 
shown  in  Fig.  51.  AB  is  the  axis  of  rev- 
olution, Cl  the  line  of  collimation,  show- 
ing only  the  object  end  to  avoid  confu- 
sion ;  the  latter  makes  with  the  axis  the 
angle  «.  Pv  C,  and  P2,  are  the  three 
points  in  a  straight  line,  the  instrument 
being  centered  over  C.  XX1  is  a  line 
drawn  perpendicular  to  the  axis  of  revo- 
lution. The  error  of  perpendicularity  is 

the  angle  /3  =  90°  —  «.     The  instrument  is  shown  with  the 
line  of  collimation  directed  toward  the  point  Pr 

If  now  the  telescope  is  revolved,  the  axis  of  revolution 
remaining  in  the  same  straight  line,  the  line  of  collimation  will 
take  the  position  CP8,  making  with  the  line  CPV  an  angle 
equal  to  2  (3.  Since  the  lens  can  not  be  moved,  the  wire  must 


X 

FIG.  51. 


112 


DIRECTION   AND  ANGLES. 


be  moved  so  that  the  line  of  collimation  will  fall  in  the  line 
CX  found  by  taking  a  point  halfway  between  P3  and  P2. 
Since  the  wire  is  nearer  the  observer  than  the  lens,  the  wire 
must  be  moved  in  a  direction  opposite  to  that  in  which  the 
forward  end  of  the  line  of  collimation  is  to  be  moved.  With 
erecting  instruments,  since  points  that  are  really  on  the  left  or 
right  appear  so,  this  rule  would  be 
followed.  With  inverting  instru- 
ments, in  which  an  object  on  the 
right  of  the  line  of  collimation  ap- 
pears to  be  on  the  left,  and  vice 
versa,  if  the  line  should  apparently 
be  moved  to  the  left,  it  should 
really  be  moved  to  the  right ;  and 
hence  the  wire  should  be  moved 
in  the  direction  in  which  it  ap- 
pears that  the  correction  should  be 
made. 

If  the  position  of  P2,  and  hence 
also  that  of  X,  is  not  known,  the 
explanation  is  as  follows  :  If,  while 
the  line  of  collimation  is  pointing 
to  P3,  a  point  is  established  there, 
and  the  instrument  is  then  turned 
on  its  vertical  axis  in  the  direction 
indicated  by  the  arrow  till  the  line 
of  collimation  again  points  to  Px, 
the  line  of  collimation  and  its  axis 
of  revolution  have  been  turned  in 
azimuth  through  an  angle  equal  to 
180°  -2/8,  and  would  be  in  the 
position  shown  at  A±  Sv  in  Fig. 
52.  The  angle  between  the  new 

u.iJ.  former  positions  of  the  axis  of  revolution  will  be  2/8.  If 
now  the  line  of  collimation  is  revolved  on  its  axis,  it  will  fall 
in  the  line  (7P4  as  much  to  the  right  of  (7P2  as  it  was  before 
to  the  left ;  and  the  angle  between  its  two  positions  pointing 
to  P3  and  P4,  will  be  4/3;  and  hence,  if  P2  had  not  been 
established,  but  only  Px,  (7,  P3,  and  P4,  the  instrument  would 


ADJUSTMENT  OF  THE  TRANSIT.  H3 

be  corrected  by  shifting  the  wire  till  the  line  of  collimation 
should  fall  on  JT2  one  fourth  the  distance  from  P4  to  Py  It 
should  be  noted  that  unless  the  axis  of  revolution  has  been 
made  horizontal,  the  points  Pv  P3,  and  P4  should  all  be  in 
about  the  same  horizontal  plane  ;  that  is,  comparatively  level 
ground  should  be  used  for  this  adjustment. 

From  the  foregoing  explanation  there  results  the  fol- 
lowing rule  for  making  this  adjustment  of  the  vertical 
wire,  or,  as  it  is  usually  called,  the  adjustment  of  the  line 
of  collimation : 

Set  the  instrument  over  a  point  in  a  comparatively  level 
stretch  of  ground.  After  leveling,  establish  a  point  about  two 
hundred  feet  in  one  direction  and  turn  the  line  of  collimation 
on  this  point,  clamping  all  motions  except  the  vertical  motion 
of  the  telescope.  Transit  the  telescope  and  set  a  point  in  the 
opposite  direction,  directly  in  the  line  of  collimation.  Loosen 
one  clamp  for  motion  in  azimuth  and  turn  the  instrument  in 
azimuth  till  the  line  of  collimation  cuts  the  first  point,  and 
clamp  all  motions  except  the  vertical  motion  of  the  telescope. 
Transit  the  telescope,  and  set  a  point  by  the  side  of  the  sec- 
ond point.  If  the  adjustment  is  perfect,  the  second  and  third 
points  will  coincide.  If  they  do  not,  move  the  vertical  wire  to 
one  side,  loosening  first  one  screw  and  then  tightening  the 
opposite,  till  the  line  of  collimation  cuts  a  point  one  fourth 
the  distance  from  the  third  point  toward  the  second.  Repeat 
the  test  for  a  check. 

It  will  usually  be  found  necessary  to  perform  the  opera- 
tion more  than  once.  All  bisections  should  be  made  with 
the  greatest  possible  precision,  using  the  clamps  and  slow- 
motion  screws  of  all  motions  except  the  vertical  motion  of  the 
telescope. 

The  horizontal  wire  will  probably  not  be  appreciably  dis- 
turbed by  this  adjustment,  but  may  be  tested  as  before.  If  it 
is  found  necessary  to  correct  it,  the  vertical  wire  must  again 
be  tested  as  above.  If  the  instrument  is  to  be  used  for 
leveling  or  for  reading  vertical  angles,  it  is  just  as  neces- 
sary as  in  the  level  that  the  horizontal  wire  be  properly 
adjusted  so  that  the  line  of  collimation  shall  be  correct  for 
all  distances. 

R'M'D  SCRV.  —  8 


114  DIRECTION   AND   ANGLES. 

IV.  The  adjustment  of  the  axis  of  revolution  of  the  tele- 
scope is  made  in  any  one  of  several  ways.  Two  will  be 
given. 

(1)  Hang  a  long  plumb  line  from  some  tall  and  firm  sup- 
port, as  a  second-story  window  sill.     Having  set  up  the  transit 
a  short  distance,  say  twenty  feet,  from  the  line,  turn  the  line  of 
collimation,  as  defined  by  the  intersection  of  the  wires,  on  a 
point  in  the  line  near  its  upper  end,  and  clamp  azimuth  motion. 
Swing  the  telescope  vertically,  noting  whether  the  intersection 
of  the  wires  remains  on  the  plumb  line.     If  not,  raise  or  lower 
one  end  of  the  axis  of  revolution  of  the  telescope  till  the  inter- 
section of  the  wires  will  follow  the  plumb  line. 

(2)  Set  up  the  instrument  near  some  tall  building  or  other 
high  object,  setting  the  intersection  of  the  wires  on  some  well- 
defined   point  near  the  top,  and   clamp   the  azimuth  motion. 
Plunge  the  telescope  downward  and  set  a  point  on  the  ground 
near  the  building  or  object.     Reverse  in  azimuth,  transit  the 
telescope,  and  set  again  on  the  point  near  the  top  and  clamp  in 
azimuth.     Drop  the  telescope,  and  see  whether  the  intersection 
of  the  wires  falls  on  the  same  point  as  before  near  the  bottom. 
If  it  does,  the  axis  is  horizontal  and  needs  no  adjustment.     If 
not,  set  a  point  midway  between  the  two  points  at  the  bottom 
and  adjust  the  axis  by  raising  or  lowering  one  end  till  the  wire 
will  cut  that  point  when  plunged  from  the  upper  point.     The 
student  will  be  able  to  make  a  diagram  showing  the  correctness 
of  these  methods  and  will  be  able  to  tell  whether  the  axis  is 
moved  in  the  second   adjustment,  so  as  to  make    the   line  of 
collimation  cut  a  point  one  fourth  the  distance  from  the  second 
to  the  first  lower  point  or  one  half  that  distance.     Unless  this 
adjustment   is   very  badly  out,  the  vertical  wire  will  not   be 
again  disturbed.     If  thought  necessary,  the  screws  may  again 
be  loosened  and  the  vertical  wire  may  be  made  truly  vertical, 
after  which   the  adjustments  for   collimation   must   again   be 
tested.     The  instrument  as  a  transit  is  now  adjusted. 

95.  Level  under  the  telescope.  To  use  the  transit  as  a 
level,  the  level  under  the  telescope  must  be  adjusted  by  the 
peg  method  as  described  for  the  leveling  instrument  in  Chap- 
ter III.,  adjusting  the  bubble,  not  the  wire. 


ADJUSTMENT   OF   THE   TRANSIT.  115 

96.  Vertical  circle.     If  the  transit  has  a  vertical  circle,  that 
should  be  adjusted  so  as  to  read  0°  when  the  bubble  under  the 
telescope  is  in  the  center  of  its  tube,  after  the  last-mentioned 
adjustment  for  leveling  has  been  made.     For  this  purpose  the 
vernier   of   the  vertical  circle   is   adjustable.      To   make   the 
adjustment,  loosen  the  small   screws   that  fasten  the  vernier 
to  the  standard  and  slide  the  vernier  along  as  is  indicated 
by  the  position  of  the  zero  of  the  circle  till  the  zeros  of  scale 
arid  vernier  coincide,  the  bubble  being  in  the  center  of  its 
tube. 

97.  Eyepiece.     After  the  wires  have   been   adjusted   they 
may  not  appear  in  the  center  of   the  field  of   view.     This  is 
because   the   eyepiece  is  not  properly  centered.     There   need 
be  no  inaccuracy  in  work  done  with  the  transit  if  this  is  not 
corrected,  but  better  seeing  will  result  if  it  is  corrected.     This 
may  be  done  by  moving  the  ring  through  which  the  eyepiece 
slides,  just  as   the  wire  ring  was  moved,   there  being  a  set 
of   screws   for  this   purpose   next  back   of   the   wire   screws. 
In  the  instrument  shown  in  Fig.  48,  this  adjustment  is  per- 
manently made.     In  such  an  instrument,  the  horizontal  wire 
may  usually  be    adjusted  with   sufficient   precision   for  ordi- 
nary work  by  merely  bringing   it   by  eye   to   the    center   of 
the  field  of  view.     The  vertical  wire  is  then  adjusted  as  de- 
scribed in  Art.  94,  III.,  and  it  is  unnecessary  to  remove  the 
telescope  from  the  standards  for  the  adjustment  of   the  line 
of   collimation.     In  inverting  instruments,  in  which  the  field 
of  view  is  limited  by  the  eyepiece  itself,  this  may  not  be  true 
unless  the  eyepiece  is  in  adjustment.     It  usually  is. 

98.  Eccentricity.     There  may  exist  errors  of  graduation  ; 
but  such  as  are  likely  to  occur  in  modern  machine-graduated 
instruments  cannot  be  detected  by  ordinary  means.     The  cen- 
ter of   the  graduated  circle  may  not  lie  in  the  axis  of   rota- 
tion, and  the  line  joining  the  zeros  of   the  verniers  may  not 
pass  through  the  center  of  the  graduated  circle.     If  the  latter 
condition  exists,  the  verniers  will  not  read  180°  apart,  except 
possibly  at   some    one  point  in  case  the  first   condition  also 
obtains.     The  second  condition  simply  means  that  the  verniers 
are  not  180°  apart,  and  no  error  will  result  from  this  cause  if 


116  DIRECTION   AND    ANGLES. 

the  same  vernier  is  read  for  both  pointings  for  the  measure- 
ment of  an  angle.  If  the  second  condition  exists  and  the  first 
does  not,  the  angular  distance  between  verniers  will  be  the 
same  for  all  parts  of  the  circle  ;  while,  if  the  first  condition 
exists,  the  angular  distance  for  different  parts  of  the  circle  will 
not  be  constant.  These  two  facts  furnish  methods  of  testing 
for  eccentricity  that  will  be  evident  to  the  student  who  is 
familiar  with  the  discussion  of  eccentricity  in  the  compass. 


THE   SOLAR  TRANSIT. 

99.  What  it  is.     The  solar  transit  consists  of  a  transit  with 
an  attachment  for  determining  the  true  meridian  by  an  obser- 
vation on  the  sun.     The  solar  transit  and  the  solar  compass, 
essentially  the    same   as   the  solar   part  of   the  solar  transit, 
have  been  extensively  used  in  laying  out  the  public  lands  of 
the  United  States.     The  solar  compass  was  invented  by  Wil- 
liam A.   Burt  of  Michigan,  and  has  become  known  as  Burt's 
solar  compass.     The  United  States  Land  Office  has  specified 
that  the  work  of  subdividing  the  public  lands  must  be  done 
with  solar  instruments  or  transits.     The  solar  compass  is  not 
now  much  used.    When  a  solar  instrument  is  used,  it  is  usually 
the  solar  transit. 

100.  Fundamental  conception.     Before  describing  the  solar 
transit,  it  will  be  necessary  to  explain  the  conceptions  on  which 
its  action  is  based.     For  this  purpose  let  the  student  imagine 
a   celestial   sphere,  concentric  with  the  earth  and   of   infinite 
radius.     This  is  not  quite  true  to  fact,  but  will  assist  the  un- 
derstanding of  the  following  statements  : 

Let  it  be  imagined  that  the  equatorial  plane  and  the  axis 
of  the  earth  are  extended  till  one  cuts  from  the  celestial  sphere 
a  circle,  called  the  celestial  equator,  and  the  other  cuts  the 
celestial  sphere  in  two  points  that  may  be  known  as  the  north 
and  south  poles.  Imagine  further  the  meridian  plane  of  the 
place  of  the  reader  extended  to  the  celestial  sphere.  It  will 
cut  from  that  sphere  a  meridian  circle.  Let  the  earth  be  con- 
ceived to  be  very  small  as  compared  with  the  celestial  sphere, 
so  that  points  on  its  surface  are  practically  at  the  center  of  the 


THE   SOLAR   TRANSIT.  117 

sphere.  If  the  reader  imagines  himself  to  be  at  the  equator, 
the  zenith  will  be  the  intersection  of  the  celestial  meridian  and 
equator.  If  he  now  imagines  that  he  moves  north,  his  zenith 
point  will  move  north  by  an  angular  amount  equal  to  the  lati- 
tude he  covers.  Moreover,  his  horizon,  which  at  the  equator 
included  the  poles,  will  be  depressed  below  the  north  pole  by 
an  equal  angular  amount.  Hence  the  altitude  of  the  north 
pole  will  at  any  place  indicate  the  latitude  of  the  place,  the 
angular  distance  from  the  pole  to  the  zenith  will  be  the  co- 
latitude  ;  the  angular  distance  zenith-equator  will  be  the  lati- 
tude, and  the  angular  distance  equator-south  horizon  will  be 
the  colatitude. 

It  is  well  known  that,  because  of  the  inclination  of  the 
ecliptic  to  the  earth's  axis,  the  sun  is  below  the  celestial 
equator  for  six  months  of  the  year  and  above  it  for  six 
months.  The  amount  that  it  is  above  or  below  it  is  con- 
stantly changing,  and  the  angular  distance  of  the  sun  from  the 
celestial  equator  at  any  moment  is  known  as  the  declination  of 
the  sun  for  that  moment.  It  is  the  same  as  terrestrial  latitude. 
Let  it  be  forgotten  for  a  time  that  the  sun  is  fixed  and  that 
it  is  the  revolution  of  the  earth  on  its  axis  that  causes  the  sun 
to  appear  to  rise  in  the  east  and  set  in  the  west,  and  let  it  be 
imagined  that  the  sun  does  the  moving  just  as  it  appears  to 
do.  Then,  if  the  sun  were  to  maintain  a  constant  declination 
for  a  whole  day,  its  path  in  the  heavens  would  correspond  to 
a  parallel  of  latitude  above  or  below  the  equator  by  the  amount 
of  the  sun's  declination  for  the  day.  On  the  21st  of  June  it 
would  correspond  to  the  extension  of  the  Tropic  of  Cancer,  and 
on  the  21st  of  December  to  the  extension  of  the  Tropic  of 
Capricorn. 

If  a  pointer  of  any  kind  should  be  directed  from  the  center 
of  the  celestial  sphere  toward  the  sun  at  any  time  during  the 
day  under  consideration,  it  would  make  with  the  equatorial 
plane  an  angle  equal  to  the  declination,  and  with  the  polar 
axis  an  angle  equal  to  the  codeclination.  If  now  this  pointer 
is  revolved  about  the  polar  axis,  keeping  the  angle  between 
them  constant,  the  pointer  will  at  all  times  point  to  some  point 
in  the  path  of  the  sun  for  the  day  ;  and  if  it  is  revolved  just 
as  fast  as  the  sun  moves,  it  will  all  day  point  to  the  sun. 


118 


DIRECTION   AND   ANGLES. 


101.  Description.  The  solar  attachment  consists  essentially 
of  an  axis  that  is  made  to  be  parallel  with  the  earth's  axis,  and 
a  line  of  sight  (pointer)  that  is  set  at  an  angle  to  the  instru- 
mental polar  axis  equal  to  the  codeclination  of  the  sun  for  the 
time  of  observation. 


FIG.  53. 


In  Fig.  53  the  polar  axis  is  marked.     It  is  made  at  right 
angles  to  the  telescope  tube,  and  hence  if  the  telescope  tube  is 


THE   SOLAR   TRANSIT.  119 

brought  into  the  plane  of  the  equator  as  shown,  and  is  also 
in  the  meridian  plane,  the  polar  axis  must  be  parallel  to  the 
terrestrial  or  celestial  polar  axis,  or,  as  is  commonly  said,  it 
must  be  pointing  to  the  pole.  If  now  the  arm  marked  AB, 
which  carries  a  line  of  sight,  is  brought  to  a  zero  reading 
on  the  declination  arc,  it  will  be  perpendicular  to  the  polar 
axis  and  practically  coincident  with  the  equatorial  plane.  If 
the  sun  is  for  the  day  on  the  equator,  the  line  of  sight,  on 
being  revolved  about  the  polar  axis,  will  cut  from  the  celestial 
sphere  the  path  of  the  sun,  or  it  can  be  at  any  time  turned 
on  the  sun.  If  the  sun  is  a  few  degrees  above  the  equator  and 
its  declination  is  set  off  on  the  declination  arc,  with  the  arc  in 
the  position  shown  in  the  figure,  and  the  line  of  sight  is  then 
revolved  about  the  polar  axis,  it  will  cut  from  the  celestial 
sphere  a  circle  parallel  to  the  equator  which  will  be  for  the 
day  the  path  of  the  sun.  So,  as  before,  the  line  of  sight  may 
be  at  any  time  turned  on  the  sun  by  simply  revolving  it  about 
the  polar  axis.  If  now  the  whole  transit  is  revolved  in  azi- 
muth so  that  the  polar  axis  no  longer  points  to  the  pole,  the 
line  of  sight  will  not,  on  being  revolved,  cut  from  the  heavens, 
the  path  of  the  sun  and  can  not  be  set  on  the  sun  at  any  time 
by  merely  revolving  it  about  the  polar  axis.  There  may  be 
one  instant  at  which  it  can  be  thus  set. 

102.  Method  of  use.  This,  then,  furnishes  the  key  to  the 
method  of  use  of  the  instrument,  which  is  as  follows  :  The 
direction  of  the  meridian  plane  being  unknown  and  desired,  set 
off  on  the  vertical  circle  of  the  instrument  the  colatitude  of  the 
place,  so  that  the  polar  axis,  when  in  the  meridian,  may  point 
to  the  pole.  Set  off  on  the  declination  arc  the  declination  for 
the  time  of  the  observation.  Now  with  the  plates  level,  so  that 
revolution  about  the  vertical  axis  may  be  only  in  azimuth, 
revolve  the  instrument  in  azimuth  and  the  line  of  sight  about 
the  polar  axis  till  the  sun  is  found  to  be  in  the  line  of  sight. 
When  this  occurs,  the  polar  axis  and  telescope  lie  in  the  merid- 
ian, and  the  instrument  may  be  clamped  and  the  line  ranged 
out.  There  is  a  small  circle  which  will  then  give  the  time  of 
day,  which  was  of  course  known  beforehand,  in  order  to  com- 
pute the  declination. 


120  DIRECTION   AND   ANGLES. 

The  declination  is  found  in  the  "  Nautical  Almanac  " J  for 
the  longitude  of  Greenwich  and  for  noon  of  each  day  in  the 
year,  and  with  the  hourly  change.  The  longitude  of  the  place 
being  known,  the  declination  for  the  time  and  place  may  be 
readily  computed.  It  must  be  remembered  that  in  most  places 
standard  time,  which  probably  is  not  the  same  as  local  time,  is 
used  and,  if  very  different  from  local  time,  allowance  must  be 
made.  A  difference  of  fifteen  minutes  will  not  ordinarily  make 
any  appreciable  error  in  the  resulting  work.  The  latitude  and 
longitude  may  be  taken  from  any  good  map,  or  the  latitude 
may  be  observed  by  measuring  on  a  previous  night  the  altitude 
of  Polaris  at  culmination,  and  adding  or  subtracting  from  the 
result  the  pole  distance  of  the  star.  See  Table  IV.,  page  364. 
It  may  also  be  observed  with  the  solar  transit,  as  will  be 
explained  later. 

The  line  of  sight  AS  consists  of  a  lens  at  A  and  a  small 
silver  disk  at  B.  The  line  of  sight  is  directed  toward  the  sun 
by  bringing  the  image  of  the  sun  formed  by  the  lens  into  the 
center  of  a  square  ruled  on  the  opposite  silver  plate.  In  order 
that  the  line  of  sight  may  be  used  when  the  sun  is  below  the 
equator,  there  is  a  lens  in  each  end,  and  a  silver  plate  opposite 
each  lens.  If  the  declination  were  south,  the  declination  arc 
would  be  reversed  from  the  position  shown  in  the  cut;  and 
the  lens  in  B  and  the  plate  in  A  would  be  used. 

103.  Limitations.  Since  there  is  a  horizontal,  that  is,  azi- 
muth, and  a  vertical  component  to  the  sun's  motion,  except  just 
at  noon,  when  the  motion  appears  to  be  all  in  azimuth,  the 
image  of  the  sun  will  appear  to  move  off  the  square  on  the 
disk,  if  it  is  left  stationary  for  a  time,  in  a  diagonal  direction, 
and  can  be  kept  in  the  square  only  by  revolving  the  line  of 
sight  about  the  polar  axis  and  shifting  the  arm  on  the  declina- 
tion arc,  the  polar  axis  being  in  the  meridian.  Just  at  noon, 
however,  the  motion  of  the  sun  is  apparently  all  horizontal,  and 
since  at  noon  the  line  of  sight  will  be  in  the  same  vertical  plane 
as  the  polar  axis  and  telescope,  the  sun's  image  will  move  out 
of  the  square  horizontally  or  between  two  of  the  lines,  so  that 

JThe  "Nautical  Almanac"  is  published  by  the  Navy  Department  at  Washing- 
ton. The  portion  relating  to  the  sun's  declination  is  published  separately  by  W.  & 
L.  E.  Gurley  and  by  G.  N.  Saegmuller,  and  perhaps  by  other  instrument  makers. 


THE   SOLAR   TRANSIT.  121 

it  could  be  kept  in  the  square  for  a  little  time  by  moving  the 
polar  axis  a  little  in  azimuth.  At  this  time,  therefore,  the 
meridian  cannot  be  correctly  determined.  It  can  not  be  well 
done  within  one  or  two  hours  either  side  of  noon. 

104.  Latitude.     For   the   same   reason,   since   at   noon  the 
exact  meridian  is  not  needed  to  get  the  sun  in  the  line  of  sight, 
this  is  the  time  to  observe  for  latitude,  as  follows  : 

Set  up  the  instrument  a  little  before  noon.  Set  off  the  co- 
latitude  (as  nearly  as  known)  on  the  vertical  circle,  and  the 
declination  for  noon  on  the  declination  arc.  Now  bring  the 
sun's  image  upon  the  silver  disk  between  the  horizontal  lines 
by  using  the  azimuth  motion  of  the  transit  and  the  vertical 
circle  tangent  screw.  The  image  will  appear  to  get  lower  as 
the  sun  goes  higher.  Keep  the  image  in  the  square  on  the 
disk  till  it  appears  to  begin  to  move  upward.  The  sun  is  then 
at  its  highest  point  ;  and  if  the  declination  has  been  properly 
set  off  and  the  plates  carefully  leveled,  the  vertical  circle  will 
read  the  colatitude  of  the  place. 

The  solar  transit  may  then  be  used  at  noon  for  finding  lati- 
tude, and  between  8  o'clock  and  10.30  o'clock  A.M.  and  1.30 
o?clock  and  5  o'clock  P.M.  for  determining  a  meridian.  It 
could  be  used  earlier  and  later  but  for  refraction,  which  is  of 
unknown  and  very  irregular  amount  near  the  horizon. 

105.  Refraction.     Thus  far  nothing  has  been  said  of  refrac- 
tion.    It  must  always  be  considered  in  setting  off  the  declina- 
tion.    The  effect  of  refraction  has  been  discussed  in  the  chapter 
on  leveling.     It  there  appears  that  the  object  is  always  seen 
higher  than  it  really  is.     Hence,  ii  the   sun's  declination  is 
north,  and  the  exact  amount  is  set  off  on  the  declination  arc, 
and  the  polar  axis  is  brought  into  the  meridian,  and  the  line  of 
sight  pointed  toward  the  sun,  the  sun's  image  would  not  be 
formed  exactly  in  the  little  square,  because  the  sun's  rays  would 
seem  to  come  from  a  higher  point  than  its  true  place.    It  is  true 
that  for  small  differences,  like  that  of  refraction  or  small  errors 
in  setting  the  latitude,  the  sun's  image  may  be  brought  on  the 
square  by  slightly  turning  the  instrument  in  azimuth,  thereby 
destroying  the  correctness  of  the  determination  of  meridian. 
Hence  it  is  necessary  to  know  and  correctly  set  off  the  latitude 


122  DIRECTION   AND   ANGLES. 

and  the  declination  corrected  for  refraction.  The  correction  for 
refraction  is  greatest  near  the  horizon,  and  is  nothing  at  the 
zenith.  Since  it  always  makes  a  luminous  body  appear  higher 
than  it  is,  the  correction  must  be  added  to  north  declinations 
and  subtracted  from  south  declinations  so  as  to  result  in  set- 
ting the  line  of  sight  to  point  higher  than  if  the  correction 
were  not  applied.  The  corrections  to  be  used  are  as  given  in 
Appendix,  Table  VII.,  page  366. 

ADJUSTMENTS  OF  THE   SOLAR   TRANSIT. 

106.  Named.     The  adjustments  of  the  solar  apparatus  are 
simple.     They  consist  in  making  the  lines  of  collimation  paral- 
lel to  each  other  and  at  right  angles  to  the  polar  axis  when  the 
declination  arc  reads  zero  ;  and  in  making  the  polar  axis  perpen- 
dicular to  the  telescope.    The  transit  is  supposed  to  be  adjusted. 

107.  Lines  of  Collimation.    These  are  made  parallel  by  mak- 
ing each  line  parallel  to  the  edges  of  the  blocks  containing 
them.    To  do  this,  remove  the  bar  carrying  the  line  of  collima- 
tion, and  replace  it  with  a  bar  called  an  adjuster,  which  is  sim- 
ply a  table  upon  which  to  rest  the  lines  of  collimation  while 
adjusting.     Rest  the  bar  containing  the  lines  of  collimation  on 
the  adjuster,  and,  by  any  means,  bring  the  sun  into  one  line  of 
collimation.     Quickly  turn  the  bar  over  (not  end  for  end).     If 
the  image  still  falls  in  the  square,  the  line  of  collimation  is  par- 
allel to  the  two  edges  of  the  blocks.     If  not,  move  the  silver 
disk  through  one  half  the  apparent  error  of  position  of  the  sun's 
image  and  try  again  till  complete.     Turn  the  bar  end  for  end, 
and  adjust  the  other  line  of  collimation.     The  two  now  being 
parallel  to  the  blocks  are  parallel  to  each  other.      Remove  the 
adjuster,  and  replace  the  bar  on  the  instrument. 

108.  Declination  vernier.     Bring  the  declination  vernier  to 
read  zero,  and,  by  any  means,  bring  the  sun  into  one  line  of 
collimation.     Quickly  and   carefully  revolve  the  bar  on   the 
polar  axis,  and  note  whether  the  sun  is  in  the  other  line  of  col- 
limation.    If  so,  the  vernier  is  in  adjustment.     If  not,  move 
the  arm  till  the  image  is  centered,  and  note  the  reading  of  the 
vernier.     Adjust  the  vernier  by  loosening  it  and  moving  it  one 
half  the  apparent  error.     Test  again. 


ADJUSTMENTS  OF   THE   SOLAR   ATTACHMENT.         123 

109.  Polar  axis.  To  make  the  polar  axis  perpendicular  to 
the  telescope  axis,  first  carefully  level  the  plates  and  the  tele- 
scope, and  then  level  the  solar  apparatus  by  the  capstan-headed 
screws  shown  underneath  the  attachment.  This  is  precisely 
the  same  as  leveling  up  an  ordinary  instrument,  and  is  per- 
formed by  the  aid  of  an  auxiliary  level  that  rests  on  the  blocks 
of  the  collimation  bar.  This  bar  is  set  so  that  the  declination 
is  zero,  and  is  brought  into  the  plane  of  the  main  telescope  and 
leveled,  then  at  right  angles  to  this  position  and  leveled  again. 
This  is  an  important  adjustment,  and  should  not  be  omitted. 


SAEGMULLER   SOLAR  ATTACHMENT. 

110.  Description  and  use.  Fig.  54  shows  another  form  of 
solar  attachment,  which  consists  simply  of  a  small  theodolite 
(so  called  because  the  telescope  will  not  transit)  attached  to 
the  top  of  the  telescope  of  an  engineer's  transit.  This  is  the 
invention  of  Mr.  George  N.  Saegmuller,  of  Washington,  D.C., 
and  is  made  by  him  and  furnished  by  other  makers  as  well. 

In  operation  it  is  similar  to  the  last-described  attachment. 
The  difference  is  that  a  telescopic  line  of  sight  is  substituted 
for  the  lens  and  disk,  and  the  small  level  is  used  in  conjunction 
with  the  vertical  circle  for  a  declination  arc. 

Suppose  the  transit  to  be  turned  with  the  object  end  of 
the  telescope  to  the  south,  and  the  telescope  level.  Assume 
north  declination.  Turn  the  object  end  of  the  telescope  down 
an  amount  equal  to  the  corrected  declination  and  bring  the 
small  bubble  to  the  center  of  its  tube.  The  angle  between 
the  main  and  small  telescopes  .then  equals  the  declination. 
If  the  object  end  is  now  pointed  to  the  equator  by  setting 
off  the  colatitude  upward  from  zero  (not  from  the  declina- 
tion reading),  the  instrument  is  set  ready  for  use. 

Turn  the  instrument  in  azimuth  and  the  small  instrument 
about  its  polar  axis  till  the  sun's  image  is  seen  in  the  small 
telescope ;  then  the  large  telescope  and  polar  axis  lie  in  the 
meridian.  There  is  a  diagonal  eyepiece  to  the  smaller  tele- 
scope to  facilitate  observations.  The  instrument  is  approxi- 
mately pointed  by  the  small  disk  sights  above  the  level  tube. 
The  objective  is  turned  up  in  setting  off  south  decimation. 


MERIDIAN   AND   TIME   BY   TRANSIT   AND   SUN.         125 

111.  Adjustments.  The  adjustments  of  this  instrument  are 
two :  the  adjustment  of  the  polar  axis,  and  that  of  the  line  of 
collimation  and  small  bubble.  The  first  is  performed,  after  the 
transit  has  been  carefully  adjusted,  by  making  the  main  tele- 
scope level,  then  leveling  the  small  telescope  over  one  set  of 
capstan-headed  screws  at  the  base  of  the  attachment  and  adjust- 
ing, and  then  over  the  other  precisely  as  in  adjusting  the  plate 
bubbles  of  the  transit,  correcting  first  by  screws  and  next  by 
tipping  the  small  telescope.  The  second  adjustment  is  per- 
formed indirectly  by  making  the  two  lines  of  collimation  par- 
allel. Measure  the  distance  between  the  centers  of  the  two 
horizontal  axes  and  draw  on  a  piece  of  paper  two  parallel 
lines  the  same  distance  apart ;  tack  this  up  at  some  distance 
from  the  instrument  and  about  on  the  same  level,  with  the 
lines  horizontal.  Make  the  two  bubbles  parallel  by  making  both 
level,  with  the  telescopes  pointing  toward  the  paper.  Set  the 
line  of  collimation  in  the  large  telescope  on  the  lower  line  on 
the  paper  and  adjust  the  wires  of  the'  small  telescope  until  the 
line  of  collimation  of  the  small  telescope  cuts  the  upper  line. 


MERIDIAN   AND  TIME  BY  TRANSIT  AND   SUN. 

112.  Meridian.  If  the  sun's  altitude  at  any  moment  is 
measured  with  the  transit,  its  azimuth  at  the  same  moment 
may  be  computed  if  its  declination  and  the  lati- 
tude of  the  place  of  observation  are  known.1 
For  in  the  spherical  triangle  pole-zenith-sun,  the 
three  sides  will  be  known  and  the  angle  Z  (the 
azimuth)  may  be  computed. 

By  Trigonometry  if  s  — 

sin  I  Z  =  \T- 


sin  b  sin  c 

from  which,  if  8  =  declination,  <j>  =  latitude,  and 

7t  =  altitude,   and    if    s'  =  ^(pole   dist.  (=90°±S)+<£  + 


cos  </>  cos  h 
1  The  same  is  true  of  any  known  star. 


(1) 


126  DIRECTION   AND   ANGLES. 

Set  up  the  transit  at  a  convenient  hour,  over  a  tack  in  a 
stake  and,  with  vernier  at  zero,  set  a  stake  some  distance  away, 
approximately  north,  and  set  the  line  of  collimation  on  this 
with  the  lower  motion.  Unclamp  the  alidade  and  turn  the 
line  of  collimation  toward  the  sun  and  measure  its  altitude, 
at  the  same  moment  clamping  the  alidade.  A  small  piece  of 
red  glass  placed  inside  the  cap  of  the  eyepiece  will  enable  the 
observer  to  look  at  the  sun;  or,  with  the  eyepiece  drawn  fully 
out,  the  image  of  sun  and  wires  may  be  received  on  a  card  held 
just  back  of  the  eyepiece.  To  observe  most  accurately,  bring 
the  horizontal  and  vertical  wires  tangent  to  the  sun's  image,  cor- 
rect the  altitude  by  the  sun's  semi-diameter,  =0°  16',  and  the 
observed  azimuth  by  16'  x  sec  h.  The  observed  altitude  must 
be  corrected  for  refraction.  The  corrections  given  in  Table  IV., 
page  364,  for  Polaris,  will  answer  if  the  column  of  latitude  is 
taken  as  altitude.  Substitute  the  corrected  altitude,  the  latitude, 
and  the  declination,  in  Equation  1,  and  solve  for  Z.  The  dif- 
ference between  the  observed  azimuth  of  the  sun  and  Z  is  the 
azimuth  of  the  line  of  stakes,  from  which  the  meridian  may  be 
laid  off.1 

113.  Time.  The  angle  at  P  in  Fig.  55  is  called  t,  the  hour 
angle.  It  is  given  by 

sin  Z  cos  h 


Reduced  to  time,  it  is  the  true  sun  time,  before  or  after  noon, 
of  the  observation.  This  must  be  corrected  by  the  equation  of 
time  (difference  between  true  sun  time  and  mean  time)  found 
in  the  "  Nautical  Almanac  "  to  give  mean  local  time.  This  must 
again  be  corrected  by  the  difference  between  mean  local  and 
standard  time.  The  result  compared  with  the  observed  time 
of  observation  will  give  the  error  of  the  watch  used. 

1  Let  the  student  show  the  error  in  azimuth  resulting  from  an  error  of  0°  01' 
in  either  latitude  or  altitude  by  computing  Z  for  values  of  <j>  differing  by  0°  01' 
and  the  same  for  values  of  h.  This  should  be  done  for  latitudes  near  30°  and 
50°  and  altitudes  of  10°  and  60°  to  show  the  effect  of  variations  in  these  quanti- 
ties. Practically  the  same  errors  arise  with  the  solar  transit  for  like  errors  of 
latitude  and  declination. 


CHAPTER   V. 

STADIA  MEASUREMENTS. 

114.  Defined.  The  stadia  is  a  device  for  reading  distances 
by  means  of  a  graduated  rod  and  auxiliary  wires  in  a  telescope. 
In  many  farm  surveys  where  the  ground  is  very  rough  and  not 
very  valuable,  the  measurements  may  be  made  with  the  stadia. 
When  so  made,  they  will  probably  be  somewhat  more  accurate 
than  if  made  with  a  chain  or  tape  with  the  care  that  would 
ordinarily  be  taken  in  the  kind  of  work  mentioned.  In  fact, 
with  the  use  of  care  and  judgment,  the  stadia  will  serve  for 
almost  all  farm  surveys. 

The  stadia  is  the  best  distance  measurer  for  extensive  topo- 
graphical surveys.  The  discussion  of  the  stadia  that  follows 
will  perhaps  indicate  its  limitations.  The  term  "stadia"  has 
been  very  loosely  used  in  this  country.  The  word  is  the  plural 
of  the  Latin  word  "stadium,"  which  means  a  standard  of  meas- 
ure. "  Stadia  "  was  the  word  adopted  by  the  Italian  engineer 
Porro  (who  invented  the  method  to  be  described)  to  indicate 
the  rod  used  by  him  in  the  application  of  his  method.  The 
English  have  retained  this  use  of  the  word  and  apply  the  word 
"  tacheometer "  to  the  transit  equipped  with  the  additional 
wires  used  in  the  method.  "  Tacheometer "  means  quick 
measurer,  and  is  applied  by  at  least  one  American  manufac- 
turing firm  to  their  high-class  instruments  that  are  equipped 
with  vertical  circle,  level  to  telescope,  and  stadia  wires.  The 
word  as  used  by  them  is  "tachymeter."  As  generally  used  in 
this  country  the  word  "  stadia  "  means  the  combination  of  in- 
strument and  rod,  but  it  is  thought  that  it  should  be  applied 
to  the  rod  only,  and  the  word  "tacheometer"  or  "tachymeter" 
is  a  very  suitable  word  to  apply  to  the  complete  transit  in- 
strument equipped  for  stadia  work.  Such  a  transit  should 

127 


128  STADIA  MEASUREMENTS. 

have  an  inverting  telescope  of  comparatively  high  power  for 
the  best  work.  The  term  "stadia  wires"  is  properly  used 
to  designate  the  wires  used  with  the  stadia. 

115.  Method  explained.  Stadia  measurements  depend  simply 
on  the  proportionality  of  the  corresponding  sides  of  similar  tri- 
angles. 

The  telescope  is  fitted  with  two  wires  in  addition  to  those 
already  described,  one  above  and  one  below  the  horizontal 
wire,  and  both  parallel  to  it.  When  a  rod  is  held  at  some 
distance  from  the  instrument,  and  the  telescope  is  properly 
focused  on  the  rod,  an  image  of  the  rod  will  be  formed  in 
the  plane  of  the  wires,  and  a  certain  definite  portion  of  this 
image  will  be  seen  between  the  two  stadia  wires.  The  rod  is 
usually  held  vertical.  If  the  telescope  is  horizontal,  the  con- 
ditions will  be  as  shown  in  Fig.  56.  The  two  wires  are  shown 
at  U  and  L.  By  drawing  lines  from  U  and  L  through  the 
optical  center  of  the  objective  0,  to  the  rod  R,  the  space  on 
the  rod  whose  image  is  included  between  the  wires  is  seen  to 
be  lu. 


FIG.  56. 

From  the  similar  triangles  OL  U  and  Olu 

7  =  T  (1) 

A  law  of  optics  is  that  the  sum  of  the  reciprocals  of  the  con- 
jugate focal  distances  of  a  convex  lens  is  equal  to  the  reciprocal 
of  the  focal  length  of  the  lens.  From  this  law,  if  /  is  the  focal 
length  of  the  objective, 

1+1=1. 

A   A  f 


SPACING  OF   THE   WIRES.  12i, 

Whence 


Ji 

g  this  value  of 
results 


Equating  this  value  of  —  with  that  obtained  from  (1),  there 


(3) 


This  is  the  distance  from  the  objective  to  the  rod.  It  is  usual 
to  require  the  distance  from  the  point  over  which  the  transit  is 
set,  to  the  rod.  If  the  distance  from  the  objective  to  the  center 
of  the  horizontal  axis,  which  is  vertically  over  the  plumb  line. 
is  represented  by  c,  the  total  distance  required  is 

V=£s  +  C/+0-  (4) 

If,  then,  a  graduated  rod  is  held  at  an  unknown  distance 
from  the  instrument,  the  distance  will  become  known  by  multi- 
plying the  space  intercepted  on  the  rod  by  the  ratio  of  /  to  i 
and  adding  to  the  product  the  constant  quantity  (/+  c). 

116.    Spacing  of  the  wires.     The  value  of  ~  must  be  known. 

Sometimes  the  wires  are  made  so  that  the  space  between  them 
is  adjustable.  This  is  not  considered  by  the  author  as  good 
practice.  The  wires  should  be  fixed.  They  are  attached  to 
the  same  ring  that  carries  the  cross  wires,  and  should  be  spaced 
at  equal  distances  from  the  horizontal  wire.  This  is  not  neces- 
sary, but  is  very  convenient  in  reading,  because  it  will  occa- 
sionally happen  that  but  one  stadia  wire  and  the  horizontal 
wire  can  be  read.  This  is  likely  to  occur  in  brushy  country, 
but  should  not  be  permitted  when  it  is  possible,  at  an  expense 
commensurate  with  the  importance  of  the  work,  to  avoid  it. 
In  case  it  should  occur,  it  is  convenient  to  have  the  wires 
equally  spaced,  because  the  reading  of  one  wire  multiplied  by 
two  will  give  a  sufficiently  accurate  determination  of  the  dis- 
tance. In  case  the  wires  are  not  equally  spaced,  separate 
values  of  the  intervals  may  be  determined  for  the  wires. 

It  is  thought  best  to  have  the  ratio  4  —  100  for  convenience. 
If  one  is  introducing  stadia  wires  in  his  transit,  he  will  find 

U'M'D  SURV.  —  9 


130  STADIA   MEASUREMENTS. 

that  it  is  practically  impossible  to  accomplish  this  result  accu- 
rately himself.  It  can  rarely  be  done  outside  of  the  maker's 
shop.  However,  it  is  not  absolutely  necessary  to  make  this 
ratio  100,  as  the  rod  may  be  graduated  to  suit  the  instrument. 

117.  To  approximate  the  value  of  /.     Focus   the   telescope 
on  a  distant  point  (theoretically  the  point  should  be  infinitely 
distant,  as  a  star)  and  measure  the  distance  from  the  center  of 
the  objective  to  the  plane  of  the  wires.     This  is  the  value  of  /. 

118.  The  value  of  c.     This  is  not  quite  constant  in  most  in- 
struments, since  the  objective  is  moved  in  and  out  in  focusing 
for  different  distances.     For  almost  all  work  done  by  the  stadia 
the  error  from  this  cause  will  not  exceed  one  tenth  of  an  inch, 
which  is  inappreciable  compared  with  the  smallest  unit  readings 
taken  by  this  method.      Single  distances  may  not  be  read  in 
careful  work  much  closer  than  the  nearest  half  foot ;  and,  in 
a  very  large  percentage  of  work  done  with  the  stadia,  the  dis- 
tances are  read  only  to  the  nearest  yard  or  meter. 

The  value  of  c  may  be  determined  by  focusing  on  an  object 
at,  say,  two  hundred  feet  distance,  and  measuring  the  distance 
from  the  objective  to  the  center  of  the  horizontal  axis.  In  some 
inverting  instruments  the  focusing  of  the  image  on  the  wires  is 
accomplished  by  moving  the  eyepiece  and  wires  together,  instead 
of  the  objective.  In  such  an  instrument  c  is  a  constant. 

119.  The  value  of  —  and  (Y+c).     If  the  value  of  -.  is  not 

known,  it  may  be  determined  as  follows  :  Select  a  strip  of  level 
ground,  drive  a  stake  and  center  it  with  a  tack.  Set  the  tran- 
sit over  this  point.  From  this  point  measure  two  distances, 
say,  one  hundred  feet  and  two  hundred  feet.  Hold  a  rod  at 
each  of  the  two  distant  points,  and  note  the  space  intercepted 
on  the  rod  at  each  point. 

(5) 

(6) 

In  each  of  these  equations,  D  and  >S  are  known,  and  hence  l£  j 
and  (/  +  <?)  may  be  found.  ^  ' 


THE   ROD. 


131 


120.  The  rod.  While  a  rod  graduated  by  lines  to  feet,  tenths 
and  hundredths,  as  the  ordinary  leveling  rod  (see  page  47), 
may  be  used  as  a  stadia,  it  is  more  convenient  to  use  a  rod  that 
is  graduated  so  as  to  give  the  distance  intercepted  by 
the  wires  directly  by  inspection,  without  reading  the 
lower  and  the  upper  wire  and  making  a  mental  sub- 
traction, as  would  be  necessary  in  the  use  of  the  level 
rod.  Various  patterns  of  rods  have  been  devised  by 
different  surveyors.  A  rod  suitable  for  a  topographic 
survey  covering  a  large  area  and  mapped  to  a  very 
small  scale  would  not  be  suitable  for  a  similar  survey 
of  a  small  tract  to  be  mapped  on  a  large  scale.  In 
the  former  case  the  units  on  the  rod  should  be  yards 
or  meters,  while  in  the  latter  case  the  units  should  be 
feet.  The  reason  for  this  difference  will  be  apparent. 
For  any  work  connected  with  land  surveys  the  rod 
should  be  graduated  to  feet,  tenths,  and  hundredths. 

An  excellent  form  of  rod  is  that  designed  by  Mr. 
T.  D.  Allin  of  Pasadena,  Cal.1  Fig.  57  is  a  draw- 
ing of  a  rod  designed  on  the  same  principle  as  that 
of  Mr.  Allin.  The  form  of  the  figures  is  somewhat 
modified,  to  the  improvement  of  the  rod.  The  essen- 
tial features  of  a  stadia  rod  are,  that  it  shall  be  well 
graduated  in  such  designs  as  will  render  it  easily  and 
quickly  read.  There  should  always  be  some  of  the 
white  rod  showing  at  every  graduation,  to  give  defi- 
niteness  to  the  position  of  the  wire.  It  will  be  noticed 
that  this  has  been  accomplished  in  the  rod  shown  in 
Fig.  57.  The  peculiarity  of  this  rod  is  that  the 
different  divisions  are  very  distinctly  marked  so  as 
to  make  the  reading  of  the  distance  very  rapid  and 

easy.     It  is  assumed  that  the  value  of  ^  is  100,  though 

this  is  not  necessary,  as  a  rod  could  be  graduated 
with  these  same  diagrams  to  fit  any  instrument.  In 
the  rod  shown,  the  distance  from  point  to  point  of  the 
finest  divisions  is  two  one-hundredths  of  a  foot,  corre- 
sponding to  two  feet  of  distance.  The  distance  from 


FIG.  57. 


1  See  "  Engineering  News,"  August  2,  1894. 


132 


STADIA   MEASUREMENTS. 


point  to  point  of  the  larger  divisions  shown  on  the  right  of 
the  rod,  is  one  tenth  of  a  foot,  corresponding  to  ten  feet  dis- 
tance. On  the  left  side  of  the  rod  the  half-foot 
points  stand  out  plainly,  corresponding  to  fifty 
feet  distance,  and  each  whole  foot  is  also  plainly 
seen,  giving  one  hundred  feet  distance.  These 
whole  feet  are  again  marked  in  sets  of  three, 
giving  three  hundred  feet  distance.  To  read 
the  rod,  set  the  upper  wire  on  some  fifty-foot 
point  and  then  run  down  the  rod  to  the  lower 
wire,  taking  in  at  once  all  the  three-hundred- 
foot  spaces,  adding  the  hundreds  and  the  final 
fifty,  then  the  tens,  and  lastly  the  single  units. 
Another  good  rod  for  long-distance  work, 
is  shown  in  Fig.  58.  The  manner  of  reading 
this  rod  is  seen  from  the  figures  at  the  side  of 
the  rod.  This  rod  is  better  suited  for  yard  or 
meter  units  than  the  Allin  rod,  and  the  author 
has  used  it  with  feet  units  with  success. 


Fio.  58. 


121.  Inclined  readings.  It  has  been  assumed 
thus  far  that  all  readings  taken  with  the  stadia 
are  horizontal.  It  will  be  evident  that  there  will  probabh- 
be  more  readings  that  are  inclined  than  horizontal.  When 
the  readings  are  inclined,  the  formulas  already  derived  must  be 
modified. 


I 


FIG.  59. 


In  Fig.  59  let  the  distance  from  I  to  A  be  required.  If  the 
rod  were  held  perpendicular  to  the  line  of  sight,  the  reading 
EF  would  at  once  give  the  inclined  distance  1C,  which  then 


INCLINED  READINGS.  133 

multiplied  by  the  cosine  of  the  angle  a  would  give  the  required 
horizontal  distance.  The  angle  a  is  determined  by  setting  the 
horizontal  cross  wire  on  a  point  on  the  rod  as  much  above  A  as 
the  center  of  the  telescope  is  above  the  ground  at  /,  and  then 
reading  the  vertical  circle  of  the  transit.  The  rod  is  usually 
held  vertical.  Therefore,  the  reading  obtained  will  be  DB.  If 
the  angles  at  E  and  F  are  considered  right  angles,  an  approxi- 
mation sufficiently  exact  for  the  purpose,  EF  is  given  by 

EF=DBcosa.  (7) 

If,  then,  R  is  the  reading  DB,  the  distance  1C  is  given  by 

1C  =  £R  cos  «  +  (/  +  C).  (8) 

But  it  is  the  distance  IGr  that  is  required. 


(9) 

From  this  equation  the  distance  is  obtained.  If  it  were  neces- 
sary to  perform  this  multiplication  for  every  sight  taken,  very 
much  of  the  usefulness  of  the  stadia  would  be  offset  by  this 
great  labor.  The  value  of  (/+<?)  varies  from,  say,  nine  inches 
to  fifteen  inches  or  more,  according  to  the  construction  of  the 
instrument.  Where  such  an  error  is  unimportant,  the  final 
term  may  be  neglected.  For  vertical  angles  between  five  de- 
grees and  six  degrees,  the  exact  angle  depending  on  the  value 
of  (/+c),  this  term  is  just  balanced  by  the  difference  between 

4  R  and  4-  R  cos2  «  ;  hence  for  angles  near  this  value  no  correc- 

tion need  be  applied,  and  the  distance  may  be  recorded  as  read. 
For  the  more  ready  computation  of  the  results,  tables  have 

been  computed  giving  the  value  vi-.R  cos2  a,  for  72  =  1,  and 

angles  from  0°  to  30°.  Such  a  table  is  Table  XIV.,  Appendix, 
pages  380-382.  To  explain  the  use  of  the  table,  suppose  a  reading 
of  1.52  feet,  usually  read  at  once  152  feet,  is  had  with  a  vertical 
angle  of  5°  30'.  Under  5°  30'  Hor.  Dist.  find  99.08,  which 
multiply  by  1.52,  and  add  the  correction  found  at  the  bottom 
of  the  page  for  c,  which  is  (/+<?)•  This  multiplication  may 


134  STADIA   MEASUREMENTS. 

be  hastened  by  the  use  of  a  slide  rule,1  which  is  an  instrument 
that  every  surveyor  should  own  and  use.  The  column  in  the 
table  headed  Diff.  Elev.  is  used  when  it  is  desired  to  know  the 
elevation  of  the  position  of  the  rod  above  the  instrument. 

122.  Difference  of  elevation.  In  Fig.  59,  since  the  instru- 
ment is  directed  to  a  point  on  the  rod  as  much  above  A  as  the 
center  of  the  instrument  is  above  the  ground  at  I  (found  by 
standing  a  rod  alongside  the  instrument),  the  difference  in 
elevation  between  the  ground  at  I  and  at  A  will  be  CGr,  and 

Ca  =  IC  sin  a 


=  v  R  cos  a  sin  a  +  (/  +  c)  sin  a 


sin2« 


For  any  reading  and  vertical  angle  the  value  of  CG  is  found 
from  the  table  in  the  same  way  as  was  the  horizontal  distance. 

123.  Diagram.  A  still  more  rapid  method  of  reducing  the 
observations  is  by  means  of  diagrams.  Omitting  the  second 
term  in  equation  (9),  the  distance  for  any  given  vertical  angle 
is  proportional  to  the  reading.  To  make  a  diagram  to  give 
the  correction  to  apply  to  the  readings  to  get  the  required  dis- 
tance, considering  only  the  first  term,  a  distance  equal  to  the 
longest  probable  reading  is  laid  off  along  one  side  of  a  sheet 
of  cross-section  paper  ruled  to  tenths  of  inches  or  other  small 
units.  Suppose  the  distance  to  be  1000  feet.  A  space  of  ten 
inches  may  then  be  laid  off,  making  the  scale  of  distance  100 
feet  to  an  inch.  Select  an  angle,  as,  say,  10°,  and  find  from 
Table  XIV.  the  true  distance  for  a  reading  of  1000  feet  and  a 
vertical  angle  of  10°.  The  difference  between  this  value  and 
1000  feet  will  be  the  correction  to  apply  to  the  reading  to  get 
the  true  distance. 

At  the  extremity  of  the  distance  line  AB,  Fig.  60,  lay  off 
at  right  angles  to  AB,  and  to  a  scale  of,  say  10  feet  to  an  inch, 
the  correction  just  found  for  the  distance  and  angle.  Sup- 
pose the  point  found  is  O.  Draw  from  C  to  A  a  straight  line. 

i  See  Chapter  VI. 


DIAGRAM. 


135 


Since  the  correction  for  this  angle  of  10°  will  be  proportional 
to  the  reading,  the  correction  for  any  other  reading  than  1000 
feet  will  be  given  by  the  ordinate  to  the  line  just  drawn  at 
the  proper  distance  from  A.  This  will  be  evident  from  the 
similarity  of  the  triangles  involved.  Thus  the  correction  for 
any  reading,  as  say  450  feet,  is  found  by  looking  along  AB  to 
the  450-foot  point,  and  taking  to  the  scale  of  10  feet  to  an  inch 


g    g    8    § 


•~  0     100    200    300    400    500    600    700    800    900    1000 
FIG.  61. 

the  ordinate  at  the  450-foot  point  to  the  10°  line.  A  similar 
line  should  be  drawn  in  a  similar  manner  for  all  other  angles. 
It  will  be  sufficiently  exact  to  draw  the  lines  for  degrees  only, 
estimating  the  minutes  by  eye. 

A  diagram  for  differences  in  elevation  may  be  made  in 
the  same  manner  (see  Fig.  61).  except  that  it  will  be  neces- 
sary to  draw  the  ten-minute  lines,  as  the  elevations  may  not  be 
estimated  with  sufficient  exactness  without  these  lines. 

It  should  be  noticed  that  the  lines  marked  1°,  2°,  3°,  etc., 
do  not  make  with  the  horizontal  line  angles  of  1°,  2°,  3°,  etc., 
but  are  drawn  to  the  points  found  by  laying  off  on  the  vertical 
the  correction  to  the  reading  for  the  given  angle,  as  found  in 
the  tables  or  as  worked  out  from  the  equations.  For  the  dif- 
ferences in  elevation,  the  differences  as  found  in  the  table  for 
the  given  distance  and  angles  are  laid  off  on  the  extreme  ver- 
tical line.  The  ten-minute  lines  may  ordinarily  be  drawn  with 
sufficient  exactness  by  dividing  the  space  between  two  adjacent 


STADIA   MEASUREMENTS. 


degree  lines  into  six  equal  parts.  These 
two  diagrams  may  be  drawn  on  one  sheet  of 
paper.1  There  are  other  forms  of  diagrams 
made  for  the  same  work.2 

The  second  term  of  the  equations  must  be 
applied  separately  when  it  may  be  necessary 
to  apply  it  at  all. 

124.  Slide  rule.    The  most  rapid  method 
of  reducing  the  differences  of  elevation  is  by 
means  of  a  special  slide  rule3  designed  for 
this  work  by  Mr.  Colby  of  St.  Louis.    A  cut 
of  this  rule  is  shown  in  Fig.  62.     It  is  about 
four  and  one  half  feet  long  by  one  and  one 
half  inches  high  by  three  inches  wide.     The 
slide  is  a  thin  strip  of  wood  with  a  varnished 
paper  scale.     Full  instructions  for  use  accom- 
pany each  rule.     With  a  little  practice  with 
this  rule,  rather  more  than  300  observations 
can  be  reduced  in  one  hour. 

125.  Graduating  a  stadia.     The  stadia  is 
not   usually   kept   in    stock    by   instrument 
makers,    probably  because    there   is   no  one 
design   that   is   generally   adopted    by   sur- 
veyors, each  surveyor  having  his  own  idea 
as  to  the  best  form.     Any  instrument  maker 
will  make  these  rods  after  a  model  pattern 
furnished  by  the  surveyor,  but  the  surveyor 
may  make  his  own  rods  very  much  cheaper 
than  he  can  get  them  made. 

Select  a  straight-grained  piece  of  any 
light  wood  that  is  riot  too  soft,  have  it  dressed 
to  the  required  length,  width,  and  thickness. 
These  dimensions  are  usually  twelve  or  fifteen 

1  Such  diagrams  are  printed  and   published  by  John 
Wiley  and  Sons,  New  York. 

2  See  "  Engineers'  Surveying  Instruments  "  by  Professor 
Ira  O.  Baker. 

3  The  principle  of  this  rule  will  be  understood  after  read- 
ing the  article  on  the  Slide  Rule  in  Chapter  VI, 


GRADUATION.  137 

feet  by  three  to  four  inches  by  seven  eighths  of  an  inch.  The 
wood  is  generally  made  uniform  in  dimensions  throughout,  but 
may  be  made  thinner  toward  the  top  and  may  be  shod  with  a 
strip  of  iron  or  brass  at  the  bottom.  Some  prefer  to  have  no  top 
or  bottom,  but  to  have  the  rod  graduated  symmetrically  from 
the  center  out  to  the  ends,  so  that  whichever  end  is  turned  up 
the  reading  is  made  equally  well.  There  seems  to  be  no  great 
advantage  in  this.  The  rod  shown  in  Fig.  57  is  three  inches 
wide  inside  of  the  side  strips. 

Give  the  rod  three  light  coats  of  white  paint,  letting  each 
coat  become  thoroughly  dry  before  the  next  is  put  on.  After 
the  final  coat  is  dry,  graduate  the  rod.  To  do  this,  set  up  the 
transit  that  is  to  be  used  with  the  rod,  on  a  level  strip  of 
ground,  and,  having  determined  the  value  (/  +  <?)'  lav  this 
distance  off  in  front  of  the  point  of  the  plumb  bob.  From  the 
point  thus  obtained,  measure  carefully  with  a  steel  tape  dis- 
tances of  one  hundred  feet,  two  hundred  feet,  three  hundred 
feet,  and  more  if  desired.  (If  the  rod  is  to  be  graduated  to 
yards  or  meters,  the  distances  should  be  one  hundred,  two 
hundred,  and  three  hundred,  of  the  units  to  be  used.)  Hold 
the  blank  rod  at  these  distances  and  carefully  mark  with  a 
hard  pencil  the  points  cut  by  the  two  wires  at  the  different 
distances. 

The  spaces  cut  should,  of  course,  be  strictly  proportional 
to  the  distances  ;  but  owing  to  inaccuracies  of  observation 
they  may  not  be  exactly  so.  Measure  the  several  spaces  inter- 
cepted and  divide  the  three-hundred  space  by  three,  the  two- 
hundred  space  by  two,  and  average  the  quotients  with  the 
one-hundred  space  to  determine  the  average  space  intercepted 
at  one  hundred  feet.  These  spaces  should  be  measured  with 
great  care,  using  either  a  good  steel  tape  or  a  standard  draughts- 
man's scale.  This  value  of  the  space  intercepted  at  one  hun- 
dred feet  having  been  determined,  it  is  divided  into  as  many 
equal  parts  as  may  be  required,  and  the  rod  is  spaced  off  for 
the  diagrams  that  have  been  selected  for  use.  The  diagrams 
are  then  laid  out  in  pencil  and  are  painted  on  the  rod,  with 
black  paint.  A  good  quality  of  paint  should  be  used,  one 
that  will  dry  with  a  dead  rather  than  a  glossy,  surface,  that  is, 
the  oil  should  be  "  killed  "  with  turpentine. 


138  STADIA   MEASUREMENTS. 

The  rod  should  then  be  allowed  to  dry.  When  it  is 
thoroughly  dry,  strips  of  hard  wood,  about  one  eighth  to  one 
fourth  of  an  inch  wider  than  the  rod  is  thick,  are  screwed  to 
the  edges  of  the  rod.  These  strips,  which  are  about  a  quarter 
of  an  inch  thick,  are  to  protect  the  graduated  surface  of  the 
rod.  They  are  very  necessary  to  a  rod  that  is  to  be  much 
used.  The  rod  should  be  shod  with  brass  or  iron  about  one 
eighth  of  an  inch  thick,  at  both  ends  if  it  is  to  be  used  either 
end  up,  and  at  the  bottom  if  it  is  to  have  a  bottom. 

For  some  kinds  of  work  reds  are  graduated  to  read  directly 
from  the  instrument  without  the  introduction  of  the  (/  +  (?) 
correction.  This  is  done  by  measuring  the  distances  one  hun- 
dred, two  hundred,  three  hundred,  or  more,  units  from  the 
plumb  bob,  noting  the  spaces  intercepted  at  these  distances, 
taking  a  mean  value  for  one  hundred  units,  and  graduat- 
ing the  rod  by  this.  The  readings  will  be  correct  on  such  a 
rod  for  but  one  distance.  If  the  rod  is  graduated  by  meas- 
uring from  the  instrument  a  distance  which  is  assumed  to  be 
about  a  mean  of  those  that  will  be  read  with  the  rod,  and  divid- 
ing the  space  intercepted  into  the  required  number  of  units,  the 
readings  taken  thereafter  will  be  correct  only  for  the  distance 
for  which  the  rod  was  graduated.  The  space  intercepted  will  al- 
ways be  that  between  the  dotted  lines  radiating  from  F,  Fig.  56. 

It  will  be  observed  that,  with  a  rod  graduated  as  first  de- 
scribed, it  is  not  necessary  that  ^-  should  be  one  hundred,  or  any 

other  definite  quantity,  since  the  actual  space  intercepted  at 
one  hundred  feet  may  be  divided  into  one  hundred  units,  and 
the  rod  may  then  be  read  as  easily  as  if  it  had  been  graduated 
to  hundredths  of  a  foot  and  'L  were  equal  to  one  hundred.  „ 

f  * 

J—  is  equal  to  one  hundred  of  the  units  used.     It  is  frequently 

convenient  to  use  a  level  rod  for  single  measurements  (as  the 
crossing  of  a  stream  on  a  survey  in  which  the  stadia  is  not  be- 
ing regularly  used),  or  to  be  able  to  graduate  rods  directly  in 
feet  and  hundredths  instead  of  taking  the  trouble  to  graduate 
them  as  has  been  described,  it  being  often  practical  to  use  such 
rods  for  a  time  as  level  rods.  For  these  reasons,  and  others, 
it  is  thought  best  to  make  i  equal  -^— . 


SMITH'S   OBSERVATIONS.  139 

126.  Smith's  observations.  It  is  possible  for  the  maker  to 
do  this  for  the  conditions  of  atmosphere  obtaining  at  the  time 
of  the  setting. 

Mr.  L.  S.  Smith,  C.  E.,  has  recently  shown1  that  system- 
atic errors  are  introduced  into  work  when  it  is  assumed  that 
a  wire  interval  determined  at  any  one  time  is  correct  for  all 
future  work.  This  is  due  to  the  fact  that  the  effect  of  refrac- 
tion is  a  variable  quantity  depending  on  the  relative  tempera- 
tures of  air  and  ground.  The  effect  of  refraction  is  very  much 
greater  near  the  ground  than  a  few  feet  above.  It  is  much 
greater  at  noon  than  before  or  after.  The  effect  varies  for 
different  distances.  It  is  also  shown  that  observers  have  dif- 
ferent "personal  equations." 

It  follows  from  the  above,  that  to  secure  the  best  results 
with  the  stadia,  and  indeed  to  avoid  cumulative  errors,  the  rod, 
if  graduated  by  the  first  method,  should  be  graduated  with  a  unit 
obtained  as  the  average  result  of  a  series  of  observations  at  vary- 
ing distances,  at  all  hours  of  the  day,  for  several  days. 

Also  if  a  wide  difference  occurs  in  climatic  or  temperature 
conditions  between  those  obtaining  when  the  rod  is  graduated 
and  those  at  the  time  of  any  survey,  the  rod  should  be  regradu- 
ated  for  the  particular  survey,  or  a  factor  applied  to  the  results. 
This  factor  should  be  determined  as  noted  below. 

It  is  also  shown  by  these  conclusions  and  the  above  result 
of  them  that  it  is  better  to  graduate  the  rod  to  standard  units 
and  depend  on  the  maker  to  place  the  wires  at  the  interval 

*g*.     It  will  be  sufficient  for  many  surveys  to  consider  this  as 

correct  and  to  use  the  reduction  tables  or  diagrams  at  once. 

But  if  close  work  is  to  be  done,  it  will  be  better  to  deter- 
mine the  wire  interval  by  observations  at  all  hours  of  the 
working  day,  and  on  several  days,  so  that  the  average  result 
shall  conform  to  the  average  to  be  expected  from  the  average 
conditions  to  prevail  during  the  survey. 

Thus  it  will  be  found  that  wires  set  ^-^  apart  will  intercept 

on  the  rod  held  at  distances  of  100  +  (/+  <?),  200  +  (/+  <?), 
etc.,  from   the   instrument,  varying   lengths,  from  possibly  a 

1  Bulletin  of  the  University  of  Wisconsin,  Engineering  Series,  Vol.  I.  No.  5. 


STADIA   MEASUREMENTS. 


little  more  than  one  unit  to  a  little  less  than  one  unit  for  each 
100  feet.  The  average  result  of  all  the  observations  would  give 
a  factor  by  which  all  observed  distances  on  the  survey  should  be 
affected  before  entering  the  reduction  tables  or  diagrams.  Thus 
suppose  the  average  intercept  for  100  +  (/  +  c)  distance  as  de- 
termined from  all  observations  should  be  1.0039,  then  all  dis- 
tances read  would  be  read  too  long,  and  should  be  multiplied  by 

before  entering  the  reduction  tables.     The  multiplica- 


tion  is  very  quickly  performed  by  the  aid  of  an  auxiliary  dia- 
gram, or  table,  or  with  the  slide  rule.  It  is  not  strictly  correct 
to  say  that  all  distances  will  be  read  too  long.  Some  will  be 
too  long  and  some  too  short,  but  the  average  result  will  be  too 
long.  By  supplying  this  factor  perhaps  no  one  of  the  distances 
is  made  correct,  but  the  errors  introduced  are  no  longer  cumu- 
lative, as  they  would  be  were  the  factor  not  used.  They  are 
now  compensating.  The  factor  should  be  determined  by  the 
person  who  is  to  make  the  observations  on  the  survey. 

By  adopting  these  precautions  and  determining  a  factor  for 
each  of  several  average  conditions,  —  say  for  each  of  the  four 
seasons,  —  a  degree  of  precision  of  one  in  two  thousand  to  one 
in  five  thousand  may  be  obtained  with  the  stadia. 

127.  Notes.  Owing  to  the  fact  that  vertical  angles  must 
be  observed  in  traversing  with  the  stadia,  the  form  of  notes 
to  be  kept  in  such  work  differs  from  that  used  with  the  transit 
and  tape  or  chain.  The  form  shown  below  is  a  good  one. 


TRAVERSE  OF  MUD  TURNPIKE,  JUNE  21,  1895. 

RIGHT   PAGE   FOR   RE- 
MARKS AND  DESCRIPTIVE 
NOTES   AND   SKETCHES. 

STATION. 
A 

AZIMUTH. 

DISTANCE, 
FEET. 

VERT. 
ANGLE. 

NEEDLE. 

37°  42' 

476 

+  4°  00' 

37°  45' 

Describe  A. 

B 

63°  26' 

324 

-  0°  26' 

63°  25' 

Describe  B. 

etc. 

etc. 

etc. 

etc. 

etc. 

etc. 

When  elevations  are  to  be  kept,  the  form  of  notes  shown  on 
page  252  is  better.  If  it  is  necessary  to  correct  the  distances, 
the  corrected  distances  may  be  written  in  red  in  the  distance 
column,  leaving  the  original  distances  for  possible  future  checks. 


CHAPTER   VI. 

LAND  SURVEY  COMPUTATIONS. 

128.  General  considerations  and  definitions.  It  will  be  evi- 
dent that,  if  the  lengths  and  bearings  of  all  the  sides  of  a 
closed  field  have  been  determined,  these  lengths  may  be  repro- 
duced to  scale  on  paper,  with  the  proper  bearings  referred  to  a 
meridian  line  drawn  on  the  paper.  It  will  be  further  evident 
that,  if  these  lines  and  bearings  have  been  correctly  determined 
and  have  been  laid  off  consecutively,  joining  end  to  end,  they 
should  form  a  closed  figure  similar  to  the  field  on  the  ground. 
A  survey  of  a  closed  field  rarely  "  closes "  on  the  drawing 
board,  owing  to  the  facts  that  distances  can  not  be  exactly 
measured,  that  the  various  sides,  offering  different  obstacles 
will  not  be  measured  with  the  same  proportionate  error,  that 
if  a  compass  is  used,  the  bearings  are  ordinarily  determined 
only  to  the  nearest  quarter  degree,  and  that  even  if  a  transit  is 
used,  these  bearings  are  not  exactly  determined.  If  the  draw- 
ing is  done  with  the  aid  of  a  protractor,  the  survey  may  close 
on  the  drawing,  even  though  there  is  some  error,  because  the 
errors  are  too  small  to  show  with  the  scale  used.  It  is  there- 
fore necessary  to  resort  to  computation  to  determine  just  what 
the  error  of  closure  is. 

To  balance  a  survey  is  to  determine  the  error  of  closure 
and,  if  it  is  not  greater  than  is  allowable,  to  distribute  a  quan- 
tity equal  to  the  error,  but  of  opposite  sign,  among  the  several 
sides,  changing  the  notes  accordingly,  so  that  the  survey  shall 
"balance "  or  close.  This  is  necessary  because  a  description  of 
a  piece  of  property  appearing  in  a  deed  ought  to  be  such  that 
it  will  indicate  a  possible  closed  tract,  and  such  descriptions 
are,  or  should  be,  made  from  surveyors'  notes.  Before  de- 
scribing the  method  of  doing  this,  it  will  be  necessary  to 
explain  a  few  terms. 

Ul 


142 


LAND   SURVEY   COMPUTATIONS. 


The  latitude  of  a  point  is  its  perpendicular  distance  from 
an  assumed  base  parallel. 

The  longitude  of  a  point  is  its  perpendicular  distance  from 
an  assumed  base  meridian. 

The  latitude  of  a  line  is  the  latitude  of  its  middle  point. 
The  longitude  of  a  line  is  the  longitude  of  its  middle  point. 
The  perpendicular  distance  of  the  final  end  of  a  given  line 
from  a  latitude  parallel  through  its  initial  end  is  the  latitude 
difference  of  the  line. 

The  perpendicular  distance  of  the  final  end  of  a  given  line 
from  the  meridian  through  its  initial  end  is  the  longitude 
difference. 

In  Fig.  63  the  latitude  difference  of  the  line  AB  is  BD ; 

the  longitude  differ- 
ence is  CB.  If  a  is 
the  bearing  of  the 
line, 

BD  =  CA  =  I  cos  a 
CB  =  I  sin  a 


FIG.  63. 


Therefore  the  lati- 
tude difference  is 
equal  to  the  length 
of  the  line  multiplied 
by  the  cosine  of  the 
bearing  ;  and  the 
longitude  difference 

equals  the  length  of  the  line  multiplied  by  the  sine  of  the  bearing. 
The  latitude  and  longitude  differences  of  the  line  BE  are 

respectively : 

BF  =  I'  cos  & 

FE  =  V  sin  ft. 

The  latitude  and  -longitude  of  the  point  jB,  referred  to  the 
meridian  and  parallel  through  A,  are  respectively  AC  and  BO. 

The  latitude  and  longitude  of  the  point  E,  referred  to  the  same 
meridian  and  parallel,  are  respectively  A  C  +  BFand  CB  +  FE. 

Latitude  differences  measured  north  are  considered  positive, 
and  those  measured  south  negative.  Longitude  differences 


ERROR  OF   CLOSURE.  143 

measured  east  are  considered  positive,  and  those  measured  west 
negative. 

North  latitudes  are  likewise  positive,  and  south  latitudes 
negative.  East  longitudes  are  positive,  and  west  longitudes 
negative. 

The  latitude  and  longitude  differences  of  the  line  JEGr  are 
respectively  —  EH  and  +  HGr.  The  latitude  and  longitude 
of  the  point  Gr  are  respectively  : 

Lat.     =  AC+BF-EH=GM, 
Long.  =  CB  +  FE  +  Ha  =  AM. 

Similarly  the  latitude  and  longitude  of  the  point  /  are 
respectively  : 

Lat.     =  AC+BF-EH-  GJ  =  -  MJ=  -  KI, 
Long.  =  CB  +FE  +  Ha-JI  =  +  AK. 

From  the  above  discussion  the  method  of  determining  the 
latitude  and  longitude  of  any  point  in  a  traverse  will  readily 
be  seen. 

The  latitude  of  any  point  in  a  traverse  is  the  latitude  of  the 
initial  point  of  the  traverse  plus  the  algebraic  sum  of  the  latitude 
differences  of  all  the  courses  of  the  traverse  to  the  given  point. 

The  student  may  formulate  a  similar  statement  for  the  lon- 
gitude of  a  point  in  a  traverse. 

In  the  figure  the  latitude  and  longitude  of  the  initial  point 
are  zero.  In  some  cases  this  may  not  be  so. 

Since  in  a  closed  traverse  the  initial  and  final  points  are 
coincident,  their  latitudes  and  longitudes  must  be  the  same, 
and  hence  the  algebraic  sum  of  the  latitude  differences  and 
the  algebraic  sum  of  the  longitude  differences  should  each 
equal  zero. 

129.  Error  of  closure.  If  the  latitude  and  longitude  dif- 
ferences of  the  courses  are  computed,  and  the  algebraic  sum  of 
each  equals  zero,  the  field  closes  ;  otherwise  it  does  not. 

If  the  field  does  not  close,  and  if  it  is  drawn  on  paper 
by  laying  off  from  a  meridian  and  parallel  the  latitudes  and 
longitudes  of  the  several  corners,  in  order,  it  will  be  found 
that  the  last  point,  which  should  be  coincident  with  the  first, 
is  some  distance  from  it.  This  distance  is  known  as  the 


144  LAND  SURVEY  COMPUTATIONS. 

error  of  closure,  and  is  the  hypotenuse  of  a  right  triangle, 
whose  two  sides  are  the  error  in  latitude  and  the  error  in  lon- 
gitude. Its  length,  therefore,  is  the  square  root  of  the  sum  of 
the  squares  of  the  errors  of  the  latitude  and  longitude.  If 
this  error  is  very  small,  or  nothing  at  all,  it  indicates  that  the 
angles  of  the  field  have  been  correctly  determined  (though  the 
bearings  may  none  of  them  be  right),  and  that  if  one  is  wrong, 
all  are  wrong  by  an  equal  amount.  It  also  indicates  that  the 
lengths  of  the  sides  have  been  measured  correctly  so  far  as 
their  relative  lengths  are  concerned,  though,  perhaps,  their 
true  lengths  have  not  been  determined.  The  correctness  of 
the  bearings  depends,  if  a  compass  has  been  used,  on  the  cor- 
rectness with  which  the  declination  has  been  determined ;  and 
the  determination  of  the  true  length  of  the  lines  depends  on 
the  length  of  the  chain  or  tape  and  the  precision  with  which  it 
has  been  handled. 

It  is  usually  assumed,  however,  that  the  constants  of  the 
compass  and  chain  have  been  correctly  determined,  and  that, 
therefore,  the  error  of  closure  is  a  true  measure  of  the  pre- 
cision of  the  survey.  If  a  transit  has  been  used  to  measure 
angles,  from  which  the  bearings  of  the  sides  have  been  com- 
puted, the  error  will  be  almost  wholly  due  to  erroneous  meas- 
urement. The  surveyor  should  know  whether  the  error  of 
closure  is  such  as  to  be  the  result  of  justifiable  or  unjustifiable 
lack  of  precision  in  the  work,  or  gross  error.  It  may  be  said 
in  this  connection  that,  in  ordinary  farm  surveying,  an  error  of 
one  in  five  hundred,  obtained  by  dividing  the  error  of  closure 
by  the  total  perimeter  of  the  field,  is  tolerable,  and  that  the 
precision  should  be,  for  good  farm  work,  as  high  as  one  in  two 
thousand,  or  better. 


BALANCING  THE  SURVEY. 

130.  Fundamental  hypotheses.  If  a  compass  is  used,  the 
method  of  correcting  the  notes  is  based  on  the  supposition  that 
any  errors  are  due  as  much  to  inaccuracy  of  angle  measure- 
ment as  to  a  lack  of  precision  in  measuring  lines.  This  does 
not  mean  that  the  error  of  closure  due  to  any  one  line  is  as 
much  owing  to  erroneous  linear  measurement  as  to  erroneous 


BALANCING   THE   SURVEY.  145 

determination  of  bearing  ;  but  that  it  is  probable  that  in  a 
line  in  which  an  error  of  bearing  would  tend  to  produce  the 
observed  error  of  closure,  such  an  error  has  been  made,  and 
little  or  no  error  in  measurement  of  length ;  while  in  a  line  in 
which  an  error  of  length  would  tend  to  produce  the  observed 
error  of  closure,  such  an  error  in  length  has  been  made  with 
little  or  no  error  in  bearing.  This  means  that  a  line  whose 
bearing  is  about  normal  to  the  line  which  represents  the  error 
of  closure,  would  be  corrected  in  bearing  and  not  in  length  ; 
while  a  line  that  is  parallel  to  the  direction  of  the  line  repre- 
senting the  error  of  closure,  would  be  corrected  in  length 
and  not  in  bearing.  Lines  whose  bearings  are  intermediate 
between  those  mentioned  would  be  corrected  in  both  bearing 
and  length,  the  greater  portion  of  the  error  being  given  to 
bearing  or  length  in  proportion  to  the  effect  that  each  would 
have  on  the  error  of  closure  ;  thus,  if  the  bearing  of  the  line 
were  nearly  normal  to  the  line  representing  the  error  of 
closure,  it  would  be  corrected  more  for  bearing  than  for 
length,  and  if  nearly  parallel  to  that  line  it  would  be  corrected 
mostly  in  length. 

The  plausibility  of  the  supposition  that  errors  of  closure 
are  due  as  much  to  errors  in  bearing  as  to  errors  in  length,  is 
clear  when  it  is  remembered  that  in  using  the  compass,  bear- 
ings are  not  read  with  greater  precision  than  to  the  nearest 
quarter  degree,  thus  making  possible  an  error  of  from  nothing 
to  seven  and  one  half  minutes.  The  error  resulting  from  an 
error  in  bearing  is  proportional  to  the  length  of  the  line  on 
which  the  error  is  made,  and  an  error  of  seven  minutes  gives  an 
erroneous  position  for  the  end  of  a  line  of  one  foot  in  five  hun- 
dred, or  as  much  as  would  be  allowed  in  measurement  of  lines. 

It  is  assumed  that  errors  in  length  will  be  proportional  to 
the  lengths  of  the  lines  measured,  although  it  is  more  probable 
that  they  are  proportional  to  the  difficulties  involved  in  the 
measurement,  and  to  the  lengths. 

131.  The  method  explained.  If  the  assumptions  are  all 
true,  it  will  follow  that  the  error  of  closure  should  be  distrib- 
uted among  the  several  sides  of  the  field  in  proportion  to  their 
several  lengths.  The  demonstration  of  this  is  as  follows  : 

R'M'D  SCRV.  — 10 


146 


LAND  SURVEY  COMPUTATIONS. 


In  Fig.  64  let  ABODE  be  the  plot  of  the  notes,  EA 
being  the  error  of  closure,  known  as  the  closing  line.  The 
line  AF  is  the  error  in  latitude,  and  EF  is  the  error  in  longi- 


FIG.  64. 


tude.  Let  the  length  of  AS  =  a,  BC=b,  CD  =  c,  DE  '=  d, 
and  let  the  sum  of  these  lengths  be  s.  Divide  EA  into  four 
parts, 

"  =  -EA,   E"E"'  =  - 

S  S 


EE'  =  -EA, 

S 


Lay  off  the  lines  BBt  ',  (7(7',  DD',  equal  and  parallel  to 
EE'.  Lay  off  also  C'C"  and  D'D",  equal  and  parallel  to 
E'E".  Lay  off  also  D"D'"  equal  and  parallel  to  E"E'". 
Connect  ^'G7".Z>'"A  This  will  be  the  correctly  closed 
field,  and  to  each  side  will  have  been  given  such  a  portion  of 
the  entire  error  of  closure,  as  that  side  is  of  the  whole  perim- 
eter. From  the  similarity  of  the  triangles  AEF,  AE'F', 


BALANCING  THE  SURVEY.  147 

AE''  F",  etc.,  it  will  be  seen  that  the  same  proportion  of  the 
error  of  latitude  and  the  error  of  longitude  has  been  given  to 
each  side,  whence  the  following  rule  : 

RULE  :  Correct  the  latitude  difference  and  longitude  differ- 
ence of  each  course  by  an  amount  determined  by  the  propor- 
tion;  the  required  correction  in  latitude  (or  longitude}  is  to  the 
total  error  in  latitude  (or  longitude)  as  the  length  of  the  course 
in  question  is  to  the  entire  perimeter  of  the  field.1 

It  will  be  observed  that  the  courses  nearly  parallel  to  HA 
have  been  corrected  mostly  in  length,  while  those  nearly 
normal  to  EA  have  been  corrected  mostly  in  bearing. 

From  the  corrected  latitude  and  longitude  differences  new 
bearings  and  lengths  are  computed. 

132.  The  practice.  This  method  is  not  strictly  followed  in 
practice.  It  is,  as  has  been  seen,  based  on  certain  assumptions 
as  to  the  probable  mode  of  occurrence  of  the  errors.  In  a 
given  case  it  may  be  that  the  surveyor  knows,  or  is  reasonably 
certain,  that  a  greater  portion  of  the  error  is  due  to  the  diffi- 
culty encountered  in  measuring  one  side,  and  in  such  a  case 
he  would  give  the  greater  portion  of  the  error  to  that  side. 
In  determining  that  this  is  true,  he  must  first  ascertain  the 
direction  of  the  closing  line  to  see  whether  the  line  supposed 
to  be  difficult  to  measure  is  parallel  or  nearly  parallel  to  the 
closing  line,  so  that  an  error  in  measurement  would  be  respon- 
sible for  the  resulting  closing  line.  It  may  be  that  it  is  believed 
that  the  error  is  not  confined  to  one  course  but  is  distributed 
over  all  the  courses,  though  not  in  the  proportion  of  their 
lengths,  as  some  may  have  offered  greater  difficulties  to  meas- 
urement than  others.  If  this  is  so,  the  error  is  distributed  in 
the  following  manner : 

RULE  :  Determine  by  judgment  which  line  has  offered  the 
least  difficulties  to  measurement  and  number  this  line  1.  Give 
to  each  of  the  other  sides  a  number  that  shall  represent  its  rela- 
tive difficulty  of  measurement,  as  2,  3,  2^,  etc.  Multiply  each 
length  by  its  number  and  find  the  sum  of  the  multiplied  lengths. 
Distribute  the  total  errors  of  latitude  and  longitude  by  the  follow- 

1  Least  square  demonstration  of  this  rule  will  be  found  in  Wright's  "Adjust- 
ment of  Observations." 


148         LAND  SURVEY  COMPUTATIONS. 

ing  proportion :  The  correction  to  the  latitude  (or  longitude) 
difference  of  any  course  is  to  the  total  error  in  latitude  (or 
longitude}  as  the  multiplied  length  of  the  course  is  to  the  sum 
of  the  multiplied  lengths. 

This  is  termed  weighting  the  courses  in  proportion  to  their 
probable  error,  and  results  in  distributing  the  whole  error  in  pro- 
portion to  the  difficulties  encountered  and  the  lengths  of  the  lines. 

When  the  lines  are  all  of  equal  difficulty  of  measurement, 
or  when  one  who  knows  nothing  of  the  field  work  balances  the 
survey,  the  first  method  should  be  used.  In  case  the  person 
making  the  survey  also  balances  it  (as  should  always  be  the 
case,  when  possible),  the  second  method  should  be  followed. 

The  original  notes  of  the  survey  should  not  be  destroyed. 
The  new  latitude  and  longitude  differences  and  lengths  and 
bearings  of  courses  may  be  written  in  the  notebook  in  red  ink 
over  the  originals.  The  originals  may  be  useful  in  the  future 
in  determining  how  the  corrections  were  made.  It  is  not  com- 
mon to  correct  the  observed  bearings  and  lengths  to  correspond 
to  the  corrected  latitude  and  longitude  differences ;  but  it  is 
well  to  do  this. 

133.  When  a  transit  is  used.  In  case  there  is  no  reason  to 
believe  there  is  any  error  in  bearings,  as  when  a  transit  is  used 
for  determining  angles,  and  therefore  the  entire  error  of  closure 
is  due  to  lack  of  uniformity  in  measurement,  the  survey  should 
be  balanced  in  such  a  manner  that  only  the  lengths  shall  be 
corrected.  No  satisfactory  rule  has  yet  been  devised  to  accom- 
plish this,  and  it  must  usually  be  done  by  trial. 

If  the  field  is  rectangular  and  one  side  may  be  taken  parallel 
to  the  meridian  of  the  survey,  or  assumed  so  for  the  purpose, 
the  following  rule  accomplishes  the  desired  result  : 

RULE  :  The  correction  in  latitude  (or  longitude}  difference 
for  any  course  is  to  the  total  error  in  latitude  (or  longitude}  as 
the  latitude  (or  longitude)  difference  of  the  course  is  to  the  arith- 
metical sum  of  the  latitude  (or  longitude)  differences. 

It  is  believed  that  the  student  can  demonstrate  the  correct- 
ness of  this  rule  for  the  assumed  conditions,  if  he  remembers 
that  the  sides  must  be  corrected  in  length  only,  their  directions 
remaining  unchanged. 


SUPPLYING   OMISSIONS.  149 

SUPPLYING  OMISSIONS. 

134.  Necessity  for.     At  times  it  is  impossible  to  measure 
the  length  of  a  line  ;  at  other  times  it  is  impracticable  to  deter- 
mine directly  its  bearing ;    and  sometimes  it  is  impossible  to 
do  either.      Moreover,  it  may  happen  that  the  length  of  one 
line  and  the  bearing  of  another  have  not  been  determined.     If 
any  one  of  these  conditions  prevails,  it  becomes  necessary  to 
supply  the  omission  in  the  notes.     To  do  this  leaves  the  work 
without  a  check,  as  all  the  errors  are  thrown  into  the  quantity 
that  is  supplied ;  hence  no  omission  that  can  in  any  way  be 
supplied  in  the  field,  should  be  permitted. 

The  cases  that  may  occur  are  the  following  : 
I.    The  length  of  one  side  may  be  wanting. 
II.    The  bearing  of  one  side  may  be  wanting. 

III.  The  bearing  and  length  of  one  side  may  be  wanting. 

IV.  The  bearing  of  one  side  and  the  length  of  another  may 

be  wanting. 

V.    The  lengths  of  two  sides  may  be  wanting. 
VI.    The  bearings  of  two  sides  may  be  wanting. 

135.  General  discussion.     It  should  be  clear  from  what  has 
preceded,  that  the  algebraic  sums  of  the  latitude  and  longitude 
differences  of   the  known  sides  are  respectively  equal  to  the 
latitude  and  longitude  differences  necessary  to  close  the  field 
(but  are  of  opposite  signs),  or  to  the  latitude  and  longitude 
differences  of  the  wanting  sides  plus  the  errors  in  latitude  and 
longitude  of  the  known  sides.     Since  there  is  no  means  of 
determining  these  errors,  the  sums  are  taken  as  the  latitude 
and  longitude  differences  of  the  defective  sides. 

It  is  therefore  possible  to  write  two  independent  equations 
as  follows  : 

I.  The  algebraic  sum  of  the  products  of  the  lengths  of  the 
defective  courses  by  the  cosines  of  their  respective  bearings 
equals   the   algebraic  sum    of   the  latitude  differences   of   the 
known  sides,  but  has  the  opposite  sign. 

II.  The  algebraic  sum  of  the  products  of  the  lengths  of  the 
defective   courses   by  the   sines   of   their   respective   bearings 
equals  the  algebraic  sum  of   the  longitude  differences  of  the 
known  sides,  but  has  the  opposite  sign. 


150  LAND  SURVEY   COMPUTATIONS. 

Having  but  two  equations,  there  can  be  but  two  unknown 
quantities,  and  these  may  be  as  in  Art.  134. 

If  Zj  and  lz  and  61  and  02  are  the  lengths  and  bearings  of 
the  defective  sides,  and  if  2  is  taken  as  a  sign  indicating  the 
algebraic  sum  of  any  series  of  quantities,  as  in  this  case  the 
latitude  and  longitude  differences,  and  L  and  M  represent  lati- 
tude and  longitude  differences  respectively,  the  above  two 
equations  may  be  written  as  follows  : 

J1cos01  +  Zacos02=-2.L,  (1) 

/!  sin  0l  +  lz  sin  02  =  -  'S.M.  (2) 

These  are,  of  course,  equivalent  to  the  following  general 
equations  : 


136.  Cases  I.,  II.,  and  III.     These  are  most  readily  solved  by 
applying  equations  (1)  and  (2).     If  but  one  quantity  is  want- 
ing, as  the  length  or  bearing  of  a  side,  either  equation  (1)  or 
equation  (2)  will  solve,  one  of  the  left-hand  terms  becoming 
part  of  the  right  member. 

If  both  the  length  and  bearing  of  one  side  are  wanting,  one 
of  the  left-hand  terms  in  e^ch  equation  becomes  part  of  the 
right  member,  and  equation  (2)  may  be  divided  by  equation 
(1),  giving 

tan  6=  ~     ,.  - 

Having  0j,  the  length  is  found  by  substituting  in  either  equation. 
The  signs  of  the  sums  of  the  known  differences  will  indi- 
cate the  direction  of  the  defective  side. 

137.  Cases  IV.,  V.,  and  VI.     The  author  thinks  the  remain- 
ing cases  are  better  solved  by  special  methods  as  follows  : 

Case  IV.  1.  The  two  imperfect  sides  are  adjacent.  With 
the  algebraic  sums  of  the  known  latitude  and  longitude  dif- 
ferences, compute  the  length  and  bearing  of  a  closing  side. 
This  side  will  form  with  the  two  defective  sides  a  triangle  in 
which  two  sides  and  one  angle  are  known.1  The  triangle  may 
then  be  solved,  giving  the  required  quantities. 

1  The  student  should  draw  a  diagram  showing  this. 


SUPPLYING   OMISSIONS. 


151 


FIG.  65. 


2.    The  defective  sides  are  not  adjacent.     Imagine  some  of 
the  sides  shifted  till  the  defective  sides  are  adjacent  and  pro- 
ceed as  before.     Thus,  in  Fig.  65, 
the   defective  sides   are   DE  and 
AB.     If  the  side  AE  is  imagined 
shifted  parallel  to  itself  to  the  po- 
sition BF,  a  closing  line  DF  may 
be    computed,  and  this  will  form 
with  ED  and  EF  (=  AB)  a  tri- 
angle. 

Case  V.  Treat  in  the  same  way 
as  Case  IV.  In  the  triangle  that 
results  there  will  be  known  the 
three  sides.  This  case  is  inde- 
terminate if  the  defective  sides  are 
parallel,  unless  the  area  is  known. 

Case  VI.  is  solved  in  the  same  way,  and  in  this  case  there  is 
known  in  the  triangle  one  side  and  all  the  angles. 

Cases  IV.  and  VI.  are  not  determinate  unless  enough  is 
known  of  the  field  to  show  which  of  two  angles  that  correspond 
to  a  given  sine  or  cosine  is  the  correct  one  to  use.  This  will 
be  evident  from  the  following  algebraic  discussion  of  Cases 
IV.,  V.,  VI. 

138.  Algebraic  solution.  Let  the  sum  of  the  known  latitude 
differences  be  denoted  by  L  and  the  sum  of  the  longitude  dif- 
ferences by  M.  Then  for  these  cases  may  be  written, 

^  cos  0j  +  ?2  cos  #2  =  ~  LI  (3) 

Zj  sin  0l  +  lz  sin  02  =  -  M.  (4) 

In  Case  IV.  let  there  be  wanting  Zx  and  #2.  Solving  (3) 
for  cosine  02  and  (4)  for  sine  02,  squaring  and  adding  and  then 
solving  for  lv  there  results 


±         -- 


-  (L  cos  0l 


It  is  evident  that  there  are  here  two  values  for  ?r  If  the  proper 
value  may  be  determined  by  a  knowledge  of  the  field,  #2  may  be 
found  by  substitution  in  either  (3)  or  (4). 

For  Case  V.  there  will  be  wanting  in  (3)  and  (4)  l^  and  1Y 


152          LAND  SURVEY  COMPUTATIONS. 

Solve  each  for  lv  equate  the  resulting  expressions,  solve  for  lz, 
and  derive 

M  cos  01  —  L  sin  0j       _  M  cos  01  —  L  sin  Ol 
2  ""  sin  01  cos  02  —  cos  6l  sin  02  ~       sin  (0X  -  02) 

from  Avhich  it  is  seen  that  the  value  is  indeterminate  when 
Bl  =  8^  or  the  defective  sides  are  parallel. 

For  Case  VI.  there  will  be  wanting  61  and  #2.  Solve  equa- 
tion (3)  for  cos^  and  equation  (2)  for  sin0r  Square  both 
resulting  equations  and  add,  getting 

7  2  _    7J2  _    71/2         7  2 

T  /I  *>»-  /I  ^1       -    -I-/       -    -if./         -    In 

L  cos  02  +  M  sm  #2  =  3  —     —  ^  —    —  2-. 

Z<2 

Let  the  right-hand  member  be  represented  by  P.     Then 


1TV1  -  cos2  02  =  P  -  i  cos  6>2; 
square,  and  solve  for  cos  02,  deriving 


from  which  it  is  evident  that  there  are  two  values,  the  correct 
one  to  be  determined  by  a  knowledge  of  the  survey.  In  the 
algebraic  solutions,  careful  attention  must  be  paid  to  the  signs 
of  the  trigonometric  functions.  Since  no  bearing  is  greater 
than  90°,  the  signs,  according  to  usual  trigonometric  concep- 
tions, would  all  be  positive  ;  but  in  equations  (3)  and  (4) 
those  signs  must  be  used  that  will  .produce  the  proper  sign  for 
the  latitude  and  longitude  differences  ;  thus,  for  a  S.E.  bearing, 
the  cosine  would  be  negative  and  the  sine  positive. 


139.  Double  longitudes.  When  the  survey  has  been  bal- 
anced, the  area  may  be  computed.  This  is  usually  done  by  the 
use  of  a  formula  involving  the  products  of  the  latitude  differ- 
ence and  double  longitude  of  each  side  of  the  survey.  The 
double  longitude  of  a  line  is  simply  twice  its  longitude,  or  the 
sum  of  the  longitudes  of  its  ends.  The  latitude  differences  are 
taken  from  the  table  of  corrected  latitude  differences  that  has 


AREAS.  153 

been  prepared  in  balancing  the  survey.  There  must  be  found 
a  convenient  method  for  determining  the  double  longitudes. 
In  Fig.  66  let  N8  be  the  reference  meridian,  the  longitude 
differences  of  the  various  sides  being  as  shown,  with  the  signs 
prefixed  to  each.  It  is  evident  that  the  double  longitudes 
of  the  first  and  last  courses  will  be  numerically  equal  to  their 
respective  longitude  differences,  but  the  sign  of  the  D.L.  of  the 
final  course  will  be  opposite  to  that  of  its  longitude  difference. 
The  D.L.  of  any  other  course,  as  BC,  is  the  sum  of  the  longi- 
tudes of  its  extremities. 

D.L.  of  BC=bB  +  cC  =  dl  +  dl  +  (-  d2). 

In  words,  the  D.L.  of  BC  is  the  D.L.  of  AB  plus  the  longi- 
tude difference  of  AB,  plus  the  longitude  difference  of  BO. 
Again,  the  D.L.  of  CD  is 


or  the  D.L.  of  CD  equals  the  D.L.  of  BC  plus  the  longitude 
difference  of  BC  plus  the  longitude  difference  of  CD. 

From  the  result  of  this  investigation  may  be  formulated  the 

RULE  :  The  double  longitude  of  any  course  is  equal  to  the 
double  longitude  of  the  preceding  course,  plus  the  longitude  differ- 
ence of  that  course,  plus  the  longitude  difference  of  the  course 

itself. 

In  applying  this  rule,  due  attention  must  be  paid  to  the 
signs  of  the  quantities.  This  rule  applies  to  the  first  course 
as  well  as  to  the  others,  if  the  preceding  course  is  considered 
to  be  zero. 

To  determine  the  D.L.  of  the  various  courses,  select  one 
course  as  the  first.  Its  longitude  difference  is  its  D.L.  To 
this  add  its  longitude  difference  and  the  longitude  difference  of 
the  second  course,  and  obtain  the  D.L.  of  the  second  course  ; 
to  this  add  the  longitude  difference  of  the  second  course  and 
the  longitude  difference  of  the  third  course,  and  obtain  the 
I).  L.  of  the  third  course,  etc.  The  correctness  of  the  work 
is  proved  if  the  D.L.  of  the  final  course  is  found  numerically 
equal  to  its  longitude  difference. 


154 


LAND   SURVEY   COMPUTATIONS. 


With  some  it  is  customary  to  select  the  most  westerly 
station  of  the  survey  as  the  point  through  which  to  pass  the 
reference  meridian.  This  practice  makes  all  of  the  D.L.'s 


FIG.  66. 


positive,  and  consequently  obviates  the  necessity  of  considering 
their  signs  in  the  subsequent  work  of  finding  areas,  and  to  this 
extent  simplifies  the  work. 

140.    To  find  the  area  of  the  field.     The   area  in  Fig.  66 
evidently  equals, 

1.  The  triangles  ABb,  Cch,  and  EeA,  plus 

2.  The  trapezoids  bBCc  and  EeDd,  minus 

3.  The  triangle  Ddh. 

Beginning  with  the  triangle  ABb  and  taking  the  triangles  and 
trapezoids  in  order  around  the  figure  to  the  right,  there  results 
the  following  equation  : 

Area  ABODE  = 
\  I  IJ>B  +  12  (bB  +  cC}  +  13  (c€-  3d)  +  Z4(Ddf  +  Ee~)+lbEe\: 


AREAS.  155 

The  third  term  is  twice  the  difference  of  the  triangles  cCh  and 
Ddh.1  This  equation  may  be  written,  paying  attention  to  the 
signs  of  the  latitude  and  longitude  differences  and  remem- 
bering that  distances  to  the  right  of  the  meridian  are  positive 
and  to  the  left  negative, 

Area  ABCDE  =  }  j  -  ^  -  12  (dl  +  dl  -  rfa)  - 
J3(2^-  d2-d,  -  df3)  +  /4  (2  (rfj-rf,)  -  d,  -  d3  -  rf4)  -  /6rf5  f . 

It  is  seen  that  the  coefficient  of  each  of  the  latitude  differences 
is  the  double  longitude  of  the  corresponding  course,  whence  the 

RULE  :  To  find  the  area  of  a  closed  survey :  Multiply  the 
latitude  difference  of  each  course  by  the  double  longitude  of  the 
course,  and  note  the  signs  of  the  products.  Divide  the  algebraic 
sum  of  these  products  by  2. 

The  sign  of  the  result  is  of  no  consequence  and  depends  on 
the  position  assumed  for  the  reference  meridian  and  the  direc- 
tion of  survey  around  the  field,  whether  clockwise  or  counter 
clockwise. 

As  stated,  the  work  is  simpler  if  the  reference  meridian 
is  chosen  through  the  most  westerly  corner. 

141.  Irregular  areas  by  offsets.  It  frequently  occurs  that 
one  side  of  a  field  that  is  to  be  surveyed  is  bounded  by  a 


FIG.  67. 

stream  or  the  shore  of  a  lake,  and  that  the  bounding  line  is 
quite  irregular  and  not  easily  run  out. 

In  Fig.   67  the   irregular   shore  line  is  a  boundary.     To 

!The  student  should  prove  this. 


156          LAND  SURVEY  COMPUTATIONS. 

determine  the  area,  two  auxiliary  courses,  AB  and  BC,  are  run 
and  used  with  the  remaining  sides  to  compute  the  area  on  their 
left.  Left  and  right  refer  to  the  direction  in  which  the  lines  are 
run.  The  additional  area  between  these  lines  and  the  shore  is 
obtained  by  measuring  offsets  normal  to  these  lines.  These 
offsets  are  measured  to  the  points  of  change  of  direction  of  the 
shore,  and  their  lengths  and  distances  from  A,  B,  or  (7,  are 
noted.  The  area  is  then  computed  by  considering  the  portions 
between  the  shore  and  base  line  and  two  adjacent  offsets  to  be 
trapezoids  or  triangles.  When  the  curve  of  the  shore  is  such 
that  the  offsets  may  be  taken  at  regular  intervals  along  a  base 
line,  the  area  is  found  by  applying  the  following  rule  : 

RULE  :  To  the  half  sum  of  the  initial  and  final  offsets  add 
the  sum  of  all  the  intermediate  offsets,  and  multiply  the  result 
by  the  common  distance  between  offsets.1 

If  the  offsets  are  taken  at  irregular  distances,  the  area  may 
be  found  as  described  in  Art.  146. 


COORDINATES. 

142.  Definitions.  It  is  becoming  common  to  call  the  lati- 
tude and  longitude  of  a  point  the  "  coordinates  "  of  the  point. 

The  base  parallel  and  meridian  are  known  as  the  "  coordi- 
nate axes,"  and  their  intersection  as  the  "  origin  of  coordinates." 
The  coordinates  of  the  origin  are  evidently  both  zero.  As 
with  latitudes  and  longitudes,  ordinates  measured  east,  or  to 
the  right  from  the  reference  meridian,  are  considered  positive; 
those  measured  west,  or  left,  are  negative;  those  measured 
north,  or  up  from  the  base  parallel,  are  positive;  and  those 
south,  or  down,  are  negative. 

It  is  not  uncommon  to  speak  of  the  latitude  ordinates  as 
"  ordinates,"  and  the  longitude  ordinates  as  "  abscissas." 

In  the  city  of  New  York,  and  in  some  other  cities,  many 
corners  to  public  property,  and  many  or  all  private  property 
corners,  are  located  by  coordinates,  the  well-established  line  of 
some  important  thoroughfare  being  taken  as  a  reference  me- 
ridian, the  origin  and  meridian  being  marked  by  monuments. 

1  The  student  will  be  able  to  show  the  truth  of  this  rule. 


COORDINATES. 


157 


The  true  meridian  through  a  well-defined  point  would  perhaps 
do  as  well,  except  that,  if  the  subdivision  of  the  city  is  on  the 
rectangular  plan,  and  not  exactly  "  with  "  the  cardinal  points,  a 
great    deal    of    com- 
putation    would     be 
avoided    by    making 
the   axes  parallel  to 
the  street  lines. 

143.  Elementary 
problems.  A  large 
amount  of  surveying 
work  is  much  facili-  x— 
tated  by  the  use  of 
coordinates,  particu- 
larly by  the  use  of 
the  methods  of  the 
two  following  prob- 
lems : 

I.  Given  the  co- 
ordinates of  two 
points,  to  find  the 
bearing  and  length  of  the  line  joining  them.1 

In  Fig.  68  the  coordinates  of  a  and  b  are  given.  The 
angle  cab  is  the  bearing  of  ab,  but  the  angle  cba  will  be  first 
found  because  the  smaller  angles  can  be  found  with  greater 
precision. 

tan  <?&«=-. 
be 

ac  =  ad  —  cd= difference  of  ordinates. 
bc=  Od—{  —  Oe)  =  difference  of  abscissas. 

Length  of  ab= =-. 

cos  cba 

To  solve  this  problem,  therefore, 

log  tan  smaller  angle  =  log  smaller  difference  of  ordinates  minus 
log  larger  difference  of  ordinates. 

(The  angle  thus  obtained  is  the  bearing  when  the  latitude  dif- 
1  The  same  as  supplying  a  wanting  course,  Art.  136. 


FIG.  68. 


158 


LAND   SURVEY   COMPUTATIONS. 


ference  is  the  greater,  and  is  the  complement  of  the  bearing 
when  the  longitude  difference  is  the  greater.) 

log  length  =  log  greater  difference  of  ordinates  minus  log  cos  of 
angle  found. 

The  letters  to  affix  to  the  bearing  may  be  determined  by 
inspection  of  the  coordinates  of  the  points. 


y 

Fia.  69. 

II.  Given  the  bearings  of  two  lines  and  the  coordinates 
of  a  point  on  each,  to  find  the  location,  i.e.  the  coordinates, 
of  the  point  of  intersection  of  the  lines.1 

The  coordinates  of  a  and  b  are  given,  and  the  bearings  of 
the  lines  as  and  br.  By  Problem  I.  find  the  bearing  of  ab  and 
the  logarithm  of  its  length.  From  the  known  bearings  of  the 
three  sides  of  the  triangle  abc,  the  three  angles  may  be  found 
and  the  triangle  solved  for  either  ac  or  be.  From  the  deter- 
mined length  of  either  be  or  ac  and  its  bearing,  its  latitude  and 
1  This  problem  is  the  same  as  supplying  two  wanting  lengths. 


COORDINATES. 


159 


longitude  differences  may  be  found  and  applied  to  the  coordi- 
nates of  a  or  6,  according  to  the  line  chosen. 

It  is  not  necessary  to  find  the  length  of  either  side  unless 
the  conditions  of  the  particular  case  require  it ;  only  the 
logarithm  need  be  found. 

144.  To  find  the  area.  Areas  are  frequently  more  readily 
computed  by  the  method  of  "coordinates." 

It  will  be  seen  in  Fig.  70  that  the  area  of  the  closed  field 


Base  Parallel  Axis  of  Longitude 
FIG.  70. 


ABGD  is  the  sum  of  the  two  areas,  IB  Cc  and  cCDd,  less  the 
sum  of  the  two  areas,  bBAa  and  aADd.  Expressed  in  an 
equation  this  becoajgs 


#3*4  +  #4*1  -  (#1*4  +  #2*1  +  #3*2  +  #4*8>  1 

(1) 

This  equation  expressed  in  words  as  a  rule  is  : 


160  LAND   SURVEY  COMPUTATIONS. 

RULE  :  To  determine  the  area  of  a  closed  field  when  the 
coordinates  of  its  corners  are  known  :  Number  the  corners  consec- 
utively around  the  field.  Multiply  each  ordinate  by  the  following 
abscissa  and  sum  the  products.  Multiply  each  ordinate  by  the 
preceding  abscissa  and  sum  the  products.  One  half  the  difference 
of  the  two  sums,  subtracting  the  second  from  the  first,  is  the  area 
of  the  field. 

Equation  (1)  may  be  written, 


8  -    \ 
This  equation  expressed  in  words  as  a  rule  is  : 

RULE  :  To  determine  the  area  of  a  closed  field  when  the 
coordinates  of  its  corners  are  known  :  Number  the  corners  consec- 
utively around  the  field.  Multiply  each  ordinate  by  the  difference 
betiveen  the  following  and  preceding  abscissas,  always  subtracting 
the  preceding  from  the  following.  One  half  the  sum  of  the  product 
is  the  area  required. 

The  second  rule  ordinarily  involves  less  work  than  the  first, 
but  there  are  certain  cases  in  which  the  first  is  used  to  better 
advantage. 

145.  To  make  the  coordinates  all  positive.  The  method  of 
determining  the  coordinates  will  perhaps  suggest  itself  to  the 
student.  To  lessen  the  danger  of  making  errors  in  signs 
it  will  be  better  to  arrange  the  axes  so  that  the  field  shall 
lie  wholly  within  the  northeast  quadrant.  All  the  coordinates 
will  then  be  positive.  This  may  be  done  as  follows  :  Deter- 
mine in  the  usual  way  the  latitudes  and  longitudes  of  the  cor- 
ners of  the  field  with  reference  to  the  meridian  and  parallel 
through  one  corner,  preferably  the  most  westerly  corner.  If 
the  most  westerly  corner  is  chosen,  the  longitudes  will  all  be 
positive  without  further  arrangement.  To  make  the  latitude 
ordinates  all  positive,  add  to  each  latitude  a  quantity  equal  to 
the  greatest  southern  latitude.  This  will  move  the  reference 
parallel  to  the  most  southern  point.  If  considered  more  con- 
venient, any  round  number,  as  100,  greater  than  the  most 
southern  latitude,  may  be  added.  An  inspection  will  ordina- 
rily be  sufficient  to  enable  the  computer  to  assume  beforehand 
proper  coordinates  for  the  first  corner. 


COORDINATES. 


101 


146.  Elongated  areas  by  offsets.  A  special  case,  in  which 
the  application  of  this  method  of  coordinates  is  advantageously 
used,  is  the  determination  of  elongated,  irregular  areas,  the 
measurements  for  which  consist  of  offsets  at  unequal  intervals 
along  a  straight  line.  In  this  case,  the  line  from  which  the 
offsets  are  measured  is  assumed  as  the  reference  parallel. 
Thus,  in  Fig.  71,  it  is  required  to  determine  the  area  between 
the  line  AH,  the  irregular  line  abcdefgh,  and  the  two  end  off- 


FIG.  71. 

sets.  The  corners  of  the  closed  field  are  AabcdefghH  and 
their  coordinates  are  as  shown  in  the  figure.  x1  and  yl  and  x2 
and  y10  are  zero.  The  equation  that  would  be  written,  follow- 
ing the  second  rule  of  Art.  144,  would  be  as  follows  : 


+  #8  o9  -  ^7)  +  y*  too  -  *8)  +  #10  to  -  %)  I  • 

yl  and  yw  being  zero,  the  first  and  last  terms  will  disappear  ; 
x1  and  x2  being  zero,  the  second  term  becomes  y^cy  and  the 
third  term  2/3£4.  What  is,  perhaps,  a  less  confusing  system  of 
writing  these  quantities  is  to  write  each  ordinate  and  its  cor- 
responding abscissa  in  the  form  of  a  fraction,  connecting  each 
ordinate  with  the  abscissas  whose  difference  is  to  be  taken  as  a 
multiplier.  Thus,  calling  the  z's  ordinates  and  the  y's  abscissas, 


The    downward    lines    to    the    right   show   the   following   or 
positive  abscissas,  and  the  downward  lines  to  the  left  show  the 
R'M'D  SURV.  —  11 


162 


LAND  SURVEY  COMPUTATIONS. 


preceding  or  negative  abscissas.  The  lines  may  be  omitted 
as  soon  as  the  student  becomes  thoroughly  familiar  with  the 
work.  This  arrangement  applies  to  any  closed  field  and  not 
alone  to  the  elongated  strip  last  described,  though  it  is  particu- 
larly applicable  to  that  method  since  the  quantities  may  be 
arranged  for  computation  as  they  are  taken  in  the  field.  Ap- 
plications of  the  coordinate  method  of  surveying  will  be  found 
in  the  problems,  pages  328-335. 

147.  Zero  azimuth.  It  is^  thought  that  the  reason  for  sug- 
gesting that  zero  azimuth  shall  be  the  north  point,  will  now  be 
clear.  Since  north  latitude  and  east  longitude  are  considered 

positive,  and  south  lati- 
tude and  west  longitude, 
negative,  a  system  of 
azimuths  should  be  so 
arranged  that  the  signs 
of  the  trigonometric 
functions  of  any  azi- 
muth shall  agree  with 
the  signs  of  the  corre- 
sponding latitude  and 
longitude  differences. 
Fig.  72  shows  the  ar- 
rangement of  signs  of 
coordinates,  and  the 
correspondence  of  the  signs  of  the  trigonometric  functions. 

There  is  thus  no  necessity  actually  to  convert  azimuth  into 
bearing  in  order  to  determine  the  signs  of  the  latitude  and 
longitude  differences,  nor  to  carry  in  mind  any  other  than  the 
ordinary  scheme  of  signs  given  in  any  work  on  Trigonometry. 
The  signs  of  the  coordinates  could,  of  course,  be  changed  to 
suit  a  south  zero  azimuth,  but  the  custom  among  all  people  to 
look  to  the  north  as  the  orienting  point,  and  the  long  use  of 
the  signs  given  for  coordinates,  seems  to  make  it  better  for  the 
surveyor  to  use  the  north  as  zero  azimuth.  Were  it  merely  a 
matter  of  changing  technical  terms  used  only  by  the  surveyor, 
such  as  changing  "  departure  "  to  "longitude  difference,"  the 
case  would  be  different. 


Cos.- 


-  72. 


DIVIDING   LAND. 


163 


DIVIDING  LAND. 

148.  Occurrence  of  the  problem.     It  sometimes  becomes  nec- 
essary to  divide  a  field  into  two  or  more  parts  of  equal  or  known 
areas.     This  occurs  when  one  man,  as  John  Jones,  sells  to  an- 
other, as  Paul  Smith,  x  acres  to  be  laid  off  in  the  northeast 
corner  of  Jones's  field.     It  also  occurs  in  the  division  of  inherited 
lands  among  the  heirs,  and  in  the  determination  of  lands  sold 
for  taxes.     When  the  taxes  are  not  paid  on  a  given  piece  of 
land,  the  land  is  sold  to  the  lowest  bidder.     This  means  that 
the  land  is  put  up  at  auction  for  the  taxes  and  expenses  of  sale, 
and  that  the  person  who  agrees  to  take  the  least  part  of  the 
whole  piece,  and  pay  therefor  the  taxes  and  expense  of  sale,  is 
given  a  title  to  that  portion  of  the  land  that  he  agrees  to  take. 
This  title  is  redeemable  by  the  original  owner  within  a  certain 
time,  specified  by  law,  after  the  expiration  of  which  time,  if  the 
title  has  not  been  redeemed,  it  becomes  vested  in  the  purchaser 
forever.     In  any  case  that  may  arise  the  original  tract  will  be 
fully  known,  either  by  previous  surveys  or  by  surveys  made  at 
the   time   and    for    the 

purpose  of  the  subdi- 
vision. It  will  also  be 
known  in  what  way  the 
land  is  to  be  divided, 
and  the  problem  then 
becomes  simply  one  of 
Geometry  or  Trigonom- 
etry. 

149.  Solution  of  the 
problem.      The  method 
of  solving  two  common 
problems  will  be  given, 
and     others     may     be 

readily  devised,  on  which  the  student  may  test  his  ingenuity. 

I.  It  is  required  to  lay  off  from  a  given  field  A  acres  by  a 
line  beginning  at  a  given  point  in  a  given  side.  Plot  the  field 
to  scale.  Let  Fig.  73  represent  the  field  so  plotted.  Let  m  be 
the  given  point.  Imagine  the  line  mD  to  have  been  run  to  the 


FIG.  73. 


164 


LAND  SURVEY  COMPUTATIONS. 


corner  nearest  the  probable  ending  point  of  the  required  line 
mg.  The  point  g  on  the  line  DE  will  first  be  determined  as 
follows  :  Consider  mBGD  as  a  closed  field  with  length  and 
bearing,  or  azimuth,  of  one  side,  mD,  wanting.  Find  bearing 
and  length  of  mD,  and  area  of  mBCD,  which  call  a.  Then 

Area  mDg  =  A  —  a. 

In  the  triangle  mDg,  the  angle  at  D  is  known,  and  the  side 
mD.  The  area  of  the  triangle  is 


A  —  a  =  |  pk  sin  D. 


Whence 


k  = 


A  — a 


^  p  sin  D 

The  triangle  may  then  be  further  solved,  giving  the  bearing 

and  length  of  mg. 

II.    It  is  required  to  lay  off  from  a  given  field  A  acres  by 

a  line  extending  in  a  given  direction. 

Let  Fig.  74  represent  the  given  plotted  field.    Select  a  corner 

of  the  field,  as  A,  such  that  from  this  corner  a  line  may  be  run 
jv  in  the  given  direction, 

cutting  off,  as  nearly  as 
may  be,  the  required 
area  A  acres.  Let  mg 
be  the  required  line, 
the  area  AFEgm  being 
equal  to  A.  If  Am 
were  known,  the  line 
mg  could  be  run.  Since 
the  length  I  should  be 
computed  for  a  check, 
and  since  it  is  some- 
what simpler  to  deter- 
mine I  first  rather  than 
Am,  I  will  be  first  de- 
termined. Imagine  the 
FIG.  74.  .  "J»  . 

line  Ah  run  in  the  given 

direction.  Consider  AFEh  a  closed  field  with  two  wanting 
lengths  —  viz.,  A h  and  Eh.  Determine  these  and  the  area 
AFEh,  which  call  a.  The  area  Amgh  =  A  —  a.  Then 


MODEL    EXAMPLES.  165 


(1)  J=p-&(tana  +  tan/3).       (2) 


«  and  /3  are  known  from  the  bearings. 


Whence  k  =  --  -~          .  (3) 

tan  a  +  tan  /3 


Substituting  in  (1), 


tan  «  +  tan  yS 


Whence         I  =  ±  V;?2  -  2  ( J.  -  a)  (tan  a  +  tan  /3) .  (4) 

Z  being  known,  k  is  determined  from  (1),  or  (3),  and  Am,  or 
hg,  from  the  small  right  triangles.  In  the  field,  find  the  point 
w,  and  run  I  on  the  given  bearing  to  its  intersection  with  DE 
at  g.  See  that  the  length  agrees  with  the  computed  length, 
and  that  gE  as  measured  agrees  with  gE  as  computed. 


MODEL  EXAMPLES. 

150.  Logarithms.  When  formerly,  in  land  surveying,  bearings  were 
read  only  to  quarter  degrees,  there  were  published  for  convenience  what 
were  known  as  traverse  tables.  These  were  nothing  more  nor  less  than 
tables  of  natural  sines  and  cosines  of  the  angles  from  0°  to  90°  for  every 
quarter-degree  multiplied  by  1,  2,  3,  4,  etc.,  to  10,  and  in  some  tables  to  100. 
The  use  of  such  a  table  made  it  unnecessary  to  multiply  the  sines  and 
cosines  of  the  bearings  to  get  the  latitude  and  longitude  differences,  since 
for  each  digit  in  the  number  expressing  the  length  of  the  course  the  differ- 
ences could  be  read  from  the  table  and  brought  to  the  right  amount  by 
moving  the  decimal  point.  The  several  quantities  were  then  added. 

With  modern  methods  of  work  the  compass  is  read  to  the  nearest  five 
minutes,  and  when  a  transit  is  used  the  angles  are  determined  to  minutes. 
Such  a  table  as  has  been  described  then  becomes  useless.  The  proper  table 
to  use  is  one  of  logarithmic  sines  and  cosines.  A  single  computation  involv- 
ing not  more  than  two  or  three  figures  can  perhaps  be  more  quickly  per- 
formed without  the  use  of  logarithms,  but  any  series  of  computations  or  a 
single  computation  involving  five  or  more  figures  can  be  more  quickly 
performed  by  logarithms.  This  does  not  mean  that  one  unaccustomed  to 
the  use  of  logarithms  can  work  with  them  so  fast  as  without,  but  a  very 
little  practice  with  them  will  in  any  case  substantiate  the  above  claim.  The 
student  should  familiarize  himself  with  a  good  set  of  logarithmic  tables.1 
The  question  will  arise  whether  four-place,  five-place,  six-place,  or  seven- 
place  tables  should  be  used.  The  decision  must  be  based  on  the  size  of  the 

:The  tables  of  Bremicker  or  Vega  are  recommended  for  work  requiring  great 
precision.  Gauss  or  Crockett  are  recommended  for  five-place  tables. 


166  LAND   SURVEY   COMPUTATIONS. 

quantities  involved  in  the  computation  and  the  precision  required  in  the 
result.  A  four-place  table  will  give  results  correct  to  three  significant 
figures  and  almost  correct  to  four  significant  figures,  probably  within  one  or 
two  units  in  the  extreme  end  of  the  table.  A  five-place  table  will  give 
results  correct  to  four  significant  figures  and  within  one  or  two  units  in  the 
end  of  the  table,  to  five  significant  figures,  and  so  on.  For  very  many  sur- 
veying computations  four-place  tables  are  good  enough,  but  for  the  general 
use  of  surveyors  five-place  tables  are  considered  better,  and  for  use  in 
connection  with  very  accurate  city  surveys,  six-place  tables  will  not  be  too 
extensive,  though  almost  all  cases  may  be  solved  properly  by  the  use  of  five- 
place  tables.  For  general  field  use  five-place  tables  are  ample.  Logarithmic 
tables  should  have  auxiliary  tables  of  proportional  parts  for  quickly  getting 
the  logarithm  of  a  number  greater  than  any  given  in  the  table  and  for 
getting  the  number  corresponding  to  a  logarithm  not  in  the  table.  Such 
tables,  with  proportional  parts  in  the  trigonometric  functions  for  tenths 
of  minutes  instead  of  for  seconds,  will  be  found  on  pages  420-464,  taken 
from  Crockett's  Trigonometry. 

151.  Example  stated.     The  notes  of  the  courses  of  a  survey  are  as 
follows : 

N.  69°  E 437.0  ft. 

S.  19°  E 236.0  ft. 

S.  27°  W 244.0ft. 

N.  71°  W 324.0  ft. 

N.  19°  W 183.5  ft. 

1424.5 

It  is  required  to  balance  the  survey  and  determine  the  area  of  the  field. 
This  example  will  be  worked  out  in  detail  as  a  model  for  the  student.  He 
is  advised  to  note  carefully  the  systematic  arrangement  of  the  work,  as  by 
such  system  much  time  is  saved.  It  is  a  case  from  practice. 

152.  Balancing.     Letting  L  represent  latitude  differences  and  M  longi- 
tude differences,  the  computation  is  arranged  as  shown  on  page  167. 

EXPLANATION.  —  We  first  write  the  logarithm  of  the  length  of  the  course, 
and  above  it  the  logarithmic  cosine  of  the  bearing,  and  below  it  the  loga- 
rithmic sine.  The  logarithm  of  L  is  then  obtained  by  adding  up,  and  the 
logarithm  of  M  by  adding  down.  The  L's  and  M's  are  then  taken  out  and 
placed  in  their  respective  columns  with  their  signs,  and  each  column  added 
algebraically,  giving  the  result  —4.9  in  L  and  +7.9  in  M.  The  error  of  clo- 
sure is  then  found.  In  the  example  given  it  is  entirely  too  large,  and  the 
field  should  be  rerun.  The  errors  in  L  and  M  are  now  distributed  among  the 
courses  in  proportion  to  the  length  of  the  sides.  It  is  not  necessary  to  be 
exact  about  this,  and  it  is  done  by  inspection.  Thus is  a  little  less  than 

one  third,  hence  the  first  corrections  are  1.5  in  L  and  2.5  in  M.  The  other 
fractions  are  treated  in  the  same  way,  the  second  being  somewhat  less  than 
one  sixth,  etc.  If  the  corrections  thus  determined  do  not  sum  up  exactly  to 


MODEL   EXAMPLES.  167 

COMPUTATIONS  FOR  THE  HARRINGTON  SURVEY. 


QUANTITY.        Loos. 

L           M 

dZ's  AND  dJf'a. 

QUANTITY.            Loos. 

L         2.19481 

+  156.6 
+  158.1 

+  405.5 
+  408.0 

\     43>      x  4  9      15 

435.2         2.63872 

cos  69°     9.55433 
437.0       2.64048 
sin  69°     9.97015 

'  1424.5  X 
x  7.9  =  2.5 

sin  68°  42'     9.96927 
405.5         2.60799 
158.1         2.19893 

M         2.61063 

tan  68°  42'    0.40906 

L         2.34858 

-223.1 
-  222.3 

+    76.5 
+    76.8 

1     "^     x  4  Q      OR 

cos  19°     9.97567 
236.0       2.37291 
sin  19°     9.61264 

'  1424.5 
X  7.9  =  1.3 

M         1.88555 

L         2.33727 

-  217.4 
-  216.6 

-  112.1 
-  110.8 

1      244         .  Q       ~  Q 

cos  27°     9.94988 
244.0       2.38739 
sin  27°     9.65705 

'  1424.5 
x  7.9  =  1.3 

M         2.04444 

L         2.02319 

+  105.5 
+  106.6 

-  308.2 
-306.4 

1     324    *49      11 

cos  71°     9.51264 
324.0      2.51055 
sin  71°     9.97567 

'  1424.5  X 
x  7.9  =  1.8 

M         2.48622 

L         2.23931 

+  173.5 
+  174.2 

-    60.7 
-    69.7 

I    183'6  -  4  9      07 

cos  19°     9.97567 
183.5       2.26364 
sin  19°     9.51264 

'  1424.5  " 
x  7.9  =  1.0 

M         1.77628 

-440.5  +484.8 
+  435.6  -476.9 

-4.9       +7.9 

Error  of  closure  =  V4.92  +  7.92  =  9.4  ±  = 


9.4     _    1 
1424.5      151 


the  respective  total  corrections,  some  one  or  more  of  them  is  slightly  altered 
to  make  the  sum  correct ;  thus,  in  the  above  example  the  corrections  first 
written  for  the  fourth  L  and  the  fifth  M  were  1.2  and  1.1  respectively,  and 
these  were  changed  to  1.1  and  1.0  to  make  the  sums  equal  4.9  and  7.9 
respectively.  The  balanced  L's  and  M's  are  now  written  in  the  column  of 
L's  and  M's  under  the  old  L's  and  over  the  old  M' s.  The  lengths  and  bear- 
ings are  now  inconsistent  with  the  balanced  L's  and  M's,  and  should  be 
corrected  to  be  consistent.  This  is  done  in  this  example  for  the  first  course 


168 


LAND   SURVEY   COMPUTATIONS. 


only.    The  tangent  of  the  bearing  is  —  ,  hence  write  the  log  M  and  subtract 

L 

the  log  L  and   get  log  tan.     Above  log  M  write  log  sin  and,  subtracting 
up,  get  log  length. 

This  completes  the  balancing. 

153.  Areas  by  latitude  differences  and  double  longitudes.  From  the 
balanced  L's  and  M's  the  double  longitudes  are  computed,  and  from  the 
L's  and  D's,  as  we  may  call  the  double  longitudes,  the  double  areas  are  com- 
puted. The  work  is  systematized  as  in  the  following  table  : 


LOGS.        DOUBLE  AREAS. 


M  of  the  First  course  is  its  D  =      405.5  2.60799 

+  M  =  +  405.5  +  L   2.19893 
+  M  =  +    75.5 


4.80692      +64110 


Second  course          D  =  +  886.5  2.94768 

+  M  =  +    75.5  -  L  2.34694 
+  M  =  -  112.1  5.29462    -  197069 


Third  course 


Fourth  course 


Fifth  course 


D  =  +  849.9  2.92937 

+  M  =  -  112.1  -  L   2.33566 
+  M  =  -  308.2 


5.26503    -  184089 


D  =  +  429.6  2.63306 

+  M  =  -  308.2  +  L  2.02776 
+  M  =  -    60.7  4.66082 


D  =  +    60.7  1.78319 

+  L  2.24105 


4.02424 


45795 


10574 


-  381158 
+  120479 


2)260679 


43560)130339.5 

2.992+  acres. 

It  is  believed  that  the  work  is  self-explanatory. 

The  results  of  the   foregoing   computations   are   usually  tabulated  in 
the  following  form : 


MODEL   EXAMPLES. 


169 


jj 

CO           CO 

g 

3 

-< 

05            CO 

I 

H 
a 

0                                            US           -# 

C5 

1 

+ 

1 

DOUBLE 

T.nwfiT- 

TUDES. 

O           O           C2           O           t— 

10            CO            05            05            0 
O          CO          •«*          CM          CD 

•^       co        co       -^i 

g 

10             0             i-H             01             t- 

0 

8 

O          >O          CM          CO          O 

O          1--          i—  i          O          CO 

0 

0 

H 
fe 

1 

+     +      1       1       1 

i—  1            CO            CO            CO            Cl 

o 

M 

0 

CO            CM            CO            CO           -t< 
O             CM             r-  1             O             1^ 

I—  1            CM             CM             r-1             rl 

0 

+      1       1      +     + 

CO               Tt<               t^ 

O5 

h 

1 

0           CO            05 

CD 

Q 

^        co 

** 

fc 

CD          CO 

co 

5 

+ 

5   *- 

co 

. 

1 

5    2 

us 
d 

CM          CM 

•^ 

Q 

H 

co                           >c       o 

CO 

^ 

+ 

co                            10        co 

>0                                      0          I- 

1 

STANCE, 

FEET. 

q       q       o       o       io 

1-^           CO           -^           -rf'           CO 

co        co       -ti        01        co 

"*          CM          Cl          CO          i—  i 

0 

1 

cs 

1-1 

j 

w     w     ^     ^     ^ 

c 

| 

05           °              °              ^            05 

C 

§ 

,    n    o    n    « 

t 

««     «j 

41 

1 1 

T3  '3 

- 


i! 

a;  u-i 
^    o 


-111 

•2^3 

n-i    -|J    ..  . 
^5     4)     O 


II 


170 


LAND   SURVEY   COMPUTATIONS. 


154.  Areas  by  coordinates.  We  shall  next  work  out  by  coordinates  the 
area  of  the  field  just  determined.  The  work  is  the  same  up  to  and  including 
the  determination  of  the  balanced  L's  and  J/'s.  An  inspection  of  these 
demonstrates  that  the  first  corner  is  the  most  westerly  corner,  and  that  the 
fourth  corner  is  the  most  southerly,  and  that  it  is  283.9  feet  south  of  the 
first.  Therefore  if  it  is  desired  to  make  all  coordinates  positive,  the  refer- 
ence meridian  will  be  passed  through  the  first  corner,  and  the  origin  of  coor- 
dinates will  be  taken  on  this  meridian  300  feet  south  of  the  first  corner.  The 
coordinates  of  the  corners  and  the  area  of  the  field  are  then  found  as  in  the 
table,  in  which  the  y's  are  the  latitude  ordinates  and  the  x's  are  the  longitude 
ordinates. 


CORNER      T 
No. 


LOGS. 


DIFF.  X's 


300.0    2.47712 

2.53757     -344.8=2-5 


2 
3 
4 
5 

+  158.2 
458.1 

-222.3 
235.8 

-  216.6 
19.2 

+  106.6 
125.8 

+  174.2 
300.0 

5.01469 

2.66096 
2.68214     +481.0  =  3-1 

405.5 
405.5 

+  75.5 
481.0 

-112.1 
368.9 

308.2 
60.7 

-60.7 
0.0 

5.34310 

2.37254 
1.56348     -  36.6  =  4-2 

3.93602 

1.28330 
2.62356     -420.3  =  5-3 

3.90686 

2.09968 
2.56691     -368.9  =  1-4 

4.66659 

X       DOUBLE  AREAS. 
SQ.  FEET. 


+  103440 


+  220341 


-     8070 


-  46408 

+  323781 

-  63108 
2)260673 

43560)130336.5 


2.992+  acres. 

Rather  more  work  is  required  by  this  method  than  by  the  method  of 
double  longitudes.  This  is  not  the  case  when  the  corners  have  been  de- 
termined by  random  lines  rather  than  by  a  continuous  traverse  around  the 
field. 

When  the  coordinates  of  the  corners  are  alone  known,  the  method  just 
given  is  by  far  the  quickest,  since  it  would  be  necessary  to  compute  the  L's 
and  M's  for  each  side  before  the  D's  could  be  obtained  or  the  areas,  just  as 
it  was  here  necessary  to  compute  the  coordinates  from  the  L's  and  M  's. 


MODEL   EXAMPLES.  171 

155.  Supplying  an  omission.  I.  In  the  example  already  used,  let  the 
bearing  and  length  of  the  second  course  be  wanting.  Adding  the  origi- 
nal £'s  and  M's,  we  get 


COURSE. 

L 

M 

1 

+  156.6 

+  408.0 

2 

3 

-  217.4 

-  110.8 

4 

+  105.5 

-  306.4 

5 

+  173.5 

-  59.7 

+  435.0 

-  476.9 

-  217.4 

+  408.0 

+  218.2 

-  67.9 

There  is  then  to  be  added  south  latitude  and  east  longitude  in  order  to 
close  the  field,  and  hence  the  line  to  be  supplied  runs  southeast. 
The  tangent  of  its  bearing  is 

LOGS. 

tan0  =  -6™  1-83187  + 

218.2  2.33885  t 

Log  tan  0  CU9302          «  =  17°  17'  +• 

Log  sin  0  9.47300  | 

Log  length  2.35887          length  =  228.5. 

The  whole  error  of  closure  being  thrown  into  this  course,  its  bearing  and 
length  have  been  materially  altered. 

II.   Let  the  lengths  of  sides  one  and  two  be  wanting.      Adding  the 
original  L's  and  M'  s,  we  get 

COURSE.  L  M 

1         +  156.6  +  408.0 
2 

3         -  217.4  -  110.8 
4 

5         +  173.5  -    59.7 

+  330.1  +  408.0 

-217.4  -  170.5 

+  112.7  +  237.5 

It  is  seen  that  southwest  bearing  must  be  that  of  the  closing  line.     Its 
bearing  and  length  are  obtained  as  in  the  last  example. 


LOGS. 

'3756 
112.7    2.05192 


tan  8  =     ™    2'37566 


Log  tan  6          0.32374  6  =  64°  37'  S.W. 

Log  sin  0          9.95591 

Log  length        2.41975  length  =  262.9. 


172 


LAND   SURVEY   COMPUTATIONS. 


We  now  have  a  triangle  composed  of  this  closing  side  and  the  two  wanting 
sides.  This  triangle1  is  formed  by  shifting  one  of  the  wanting  sides  par- 
allel to  itself.  In  this  triangle  there  are  known  the  bearing  and  length  of 
the  closing  side  just  found  and  the  bearings  of  the  two  other  sides,  and 
hence  there  are  known  all  the  angles  and  one  side. 

If  the  apexes  are  lettered  A,  B,  C,  and  the  sides  opposite  the  apexes 
a,  b,  and  c.  respectively,  and  if  a  is  the  closing  side  just  found,  b  side  2,  and 
c  side  4,  we  have  from  the  known  bearings 


Whence 


a  sin  B     262.9  sin  44°  24' 


Angle  A  =  52°  00' 

"      B  =  44°  23' 

"      C  =  83°  37' 

180°  00' 


Solve  for  course  4.     The  other  problems  are  similarly  solved. 


Log  262 
Log  sin 

Log  sin 
Log  b 

sin  A 
.9 
44°  23' 

52°  00' 

sin  52°  00' 
2.41975 
9.84476 

2.26451 
9.89653 
2.36798 

b  =  233.3. 

THE   PLANIMETER. 


156.  Description.  The  most  elegant  and  rapid  method  to 
obtain  the  area  of  an  irregular  figure  is  to  draw  the  figure  to 
scale  and  measure  the  area  with  a  planimeter.  There  are  three 
kinds  of  planimeters,  shown  in  Figs.  75,  76,  and  77. 


FIG.  75. 


Fig.  75  shows  the  polar  planimeter,  the  most  commonly 
used.  Fig.  76  is  a  suspended  planimeter,  which  is  a  polar 
planimeter  so  arranged  that  the  wheel  <?,  Fig.  75,  will  roll  on 
a  polished  surface  instead  of  on  the  drawing.  In  Fig.  76  the 
axle  of  the  wheel  is  turned  by  contact  with  a  surface.  This 
instrument  is  more  accurate  in  its  results  than  the  polar  planim- 
eter. Fig.  77  is  a  rolling  planimeter.  This  is  the  most  costly 
and  the  most  accurate  of  the  three  forms.  Its  principle  of 
action  is  somewhat  different  from  that  of  the  polar  planimeter. 

1  The  student  should  draw  the  triangle. 


THE   PLANIMETER. 


173 


The  polar  planimeter,  the  one  most  used,  consists  of  two 
arms,  Ti  and  /,  one  of  which,  /,  is  of  fixed  length,  and  the  other 
is  adjustable  through  the  frame  shown  on  the  left.  There  is  a 


FIG.  76. 


clamp  back  of  the  point  g  and  a  slow-motion  screw/  for  setting 
the  adjustable  arm  to  the  required  length.  The  arm  j  is  piv- 
oted at  &,  and  directly  over  &,  on  the  frame,  is  a  single  gradua- 
tion which  indicates  the  length  of  the  adjustable  arm.  There 


is  also  a  wheel  c,  whose  axis  of  revolu- 
tion  is  parallel  to  the  arm  A,  and,  in  the  dis-  ^ 
cussion  that  follows,  will  be  considered  a  part  of 
that  arm.     The  wheel  is  so  mounted  as  to  be  almost 
frictionless.     The  disk  I  records  entire  revolutions  of 
the  wheel,  while  by  the  vernier  m  the  fractional  rev 
olutions  are  read. 

157.    Use.     The  point  e  is  fixed  in  the  drawing  board  pref- 
erably outside  of  the  figure  to  be  measured.     The  tracer  d  is 


174  LAND   SURVEY  COMPUTATIONS. 

then  placed  on  a  point  in  the  circumference  of  the  figure, 
and  the  wheel  c  is  read.  The  wheel  could  be  set  to  zero,  but 
this  is  not  easy  to  do,  and  hence  it  is  read  at  whatever  it  hap- 
pens to  be.  The  tracer  d  is  now  moved  carefully  around  the 
circumference  and  stopped  at  the  beginning  point.  The  wheel 
is  again  read,  and  the  difference  of  the  two  readings  indicates 
the  number  of  revolutions  made  by  the  wheel.  The  instru- 
ment is  so  made  that  if  the  tracer  is  moved  clockwise,  the 
wheel  will  roll  in  the  direction  of  its  graduation,  and  hence 
the  final  reading  will  be  greater  than  the  initial  reading,  and 
vice  versa.  The  wheel  by  its  vernier  reads  to  thousandths  of 
a  revolution.  The  principle  of  the  instrument  is  such  that 
the  distance  rolled  by  the  wheel,  or  the  number  of  revolutions 
times  the  circumference  in  inches,  multiplied  by  the  length  of 
the  arm  in  inches,  is  the  area  in  square  inches  bounded  by  the 
path  of  the  tracer  d.  The  arm  h  is  generally  so  set  that  ten 
times  the  number  of  revolutions  is  the  area.  If  other  units,  as 
tenths  of  a  foot,  are  used  for  the  above-named  linear  units,  the 
area  will  be  given  in  square  units  of  the  same  kind.  In  any 
event,  it  is  the  area  of  the  drawing  that  is  measured.  If  this 
drawing  has  been  made  to  a  scale  of  say  forty  feet  to  an  inch, 
a  square  inch  of  paper  is  equivalent  to  sixteen  hundred  square 
feet,  and  the  area  given  by  the  instrument  must  be  multiplied 
by  this  quantity  to  get  the  area  in  square  feet.  If  the  fixed 
point  is  placed  inside  of  the  area  to  be  measured  so  that  the 
tracer  d  in  circumscribing  the  required  area  makes  a  complete 
revolution  about  e,  there  must  be  added  to 
the  result  in  square  inches  a  certain  con- 
stant area  —  a  constant  for  each  instru- 
ment—  called  the  area  of  the  zero  circum- 
ference. 


158.  Theory.  The  following  is  a  dis- 
cussion  of  the  theory  of  the  instrument, 
written  for  those  who  are  not  familiar  with 
the  principles  of  the  Calculus. 

In  the  figures  that  follow,  the  essential 
FIG.  78.  parts  of  the  instrument  are  lettered  as  in 

Fig.  75.     The  instrument  is  so  constructed  that  neither  d  nor 


THE   PLANIMETER.  175 

c  can  cross  j.  In  circumscribing  an  area,  the  curved  path  of  d 
may  be  conceived  as  divided  into  an  infinite  number  of  infini- 
tesimal portions,  each  a  straight  line. 

Each  of  the  small  portions  may  be  conceived  as  made  up  of 
two  parts  or  component  motions,  one  radial  with  reference  to 
e,  and  the  other  circumferential. 

With  the  radial  motion  the  value  of  the  angle  </>  is  constantly 
changing,  while  with  the  circumferential  motion  the  value  of  <f> 
and  the  length  of  the  line  ed  remains  fixed.  In  Fig.  78  the 
component  motions  sd  and  sdf,  and  their  resultant  motion  dd', 
are  shown  greatly  enlarged.  It  will  be  evident,  that,  in  cir- 
cumscribing a  closed  figure,  each  minute  movement  of  d  toward 
e  will  have  its  corresponding  movement  from  e  with  the  same 
value  of  (f>.  Each  element  of  right-hand  circumferential  motion 
will  have  its  corresponding  element  of  left-hand  circumferential 
motion  ;  but,  since  d  will  be  farther  from  e  for  one  than  for  the 
other,  these  corresponding  elements  will 
not  be  made  with  equal  values  of  <j>. 

When  the  plane  of  the  wheel  <?, 
Fig.  79,  passes  through  e,  the  angle  dee 
tis  a  right  angle,  and,  if  d  is  revolved 
about  e  with  <£  constant,  there  will  be  no 
rolling  of  the  wheel  <?,  because  the  direc- 
tion of  motion  of  all  points  of  the  in- 
strument about  e  is  circumferential,  and 
the  radius  of  motion  of  the  point  c  is 
ec,  and  this  is  normal  to  the  axis  of  the 
wheel,  and  hence  the  motion  of  the  wheel  FlG'  79' 

is  parallel  to  its  axis,  and  the  wheel  simply  slips  and  makes 
no  record.  The  path  described  by  d  for  the  particular  value 
of  <f>  that  brings  about  the  above  result,  is  known  as  the  zero 
circumference.  Its  radius,  ed,  may  be  easily  shown  to  be 


(1) 
(2) 


It  will  be  evident  that  if  d  could  be  moved  outward  till  c 
should  fall  in  the   line  ek,  and   then   rotated   clockwise,  the 


176 


LAND   SURVEY  COMPUTATIONS. 


motion  of  c  would  be  all  rolling  motion,  and  would  be,  looking 
from  c  to  J,  clockwise.  The  wheel  is  graduated  so  as  to  record 
positively  for  this  kind  of  motion.  Between  these  two  posi- 
tions the  motion  of  the  wheel  will  be  partly  slip  and  partly  roll, 
the  amount  of  each  depending  on  the  value  of  <£;  and  the  roll 
will  all  be  clockwise.  It  will  be  further  evident  that,  if  d 
were  moved  in  till  c  should  fall  in  the  line  ke  produced,  and 
then  rotated  clockwise,  the  motion  of  c  would  be  all  roll  and 
would  be  counter-clockwise  or  left-handed,  when  looking  as 
before  from  c  to  d.  For  d  between  the  zero  circumference  and 
the  last-named  position,  clockwise  motion  will  produce  a  motion 
of  c  partly  roll  and  partly  slip.  The  amount  of  each  will  de- 
pend on  the  value  of  <£,  and  the  roll  will  be  counter-clockwise. 
Hence,  clockwise  motion  of  c  will  be  caused  by  positive  motion 
of  d  outside  the  zero  circumference,  and  by  negative  motion  of 
d  inside  the  zero  circumference,  and  vice  versa;  arid,  since  the 
amount  of  roll  for  a  given  motion  of  the  tracer  depends  on  the 
value  of  <£,  any  two  equal  infinitesimal  motions  in  opposite 
directions  with  the  same  value  of  <£  will  produce  no  resultant 
roll  of  the  wheel,  while,  if  made  with  unequal  values  of  </>, 
there  will  be  a  resultant  roll  of  the 
wheel  that  can  be  read.  For  these 
reasons,  the  radial  components  in  cir- 
cumscribing a  closed  figure  cause  no 
resultant  rolling  of  the  wheel  and  may 
be  neglected,  while  the  circumferential 
components  do  cause  a  resultant  roll 
and  must  be  considered.  It  will  be 
shown  that  the  roll  of  the  wheel  for  a 
given  circumferential  motion  of  d  is 
proportional  to  the  area  included  be- 
tween the  path  of  d,  the  radial  lines 
from  e  to  the  initial  and  final  points  of 
rf's  path,  and  the  arc  of  the  zero  circumference  included  between 
those  lines. 

Let  dd',  Fig.  80,  be  a  minute  circumferential  component  of 
<Fs  motion  due  to  the  movement  of  the  instrument  about  e  as  a 
center,  through  the  angle  A,  </>  remaining  constant.  The  wheel 
will  move  through  the  arc  cc',  and  will  partly  roll  and  partly 


THE   PLANIMETER.  177 

slip.  The  rolling  component  of  its  motion  will,  of  course,  be 
normal  to  its  axis,  and  may  be  represented  by  the  line  <?«,  esc' 
being  considered  an  infinitesimal  right-angled  triangle.  Let  A 
be  the  length  in  linear  units  of  an  arc  of  radius  unity  and  cen- 
tral angle  A.  (For  any  other  radius  X  the  length  of  the  arc 
for  a  central  angle  of  A  would  be  X  times  A.}  The  angle  cec' 
is  A,  and  the  arc  cc'  is  given  by 

cc'  =J~e.A. 
The  roll  of  the  wheel  is 

cs   =  c'e  •  A  -  cos  c'cs. 

The  angle  A  being  very  small,  cc'  may  be  considered  perpen- 
dicular to  c'e,  whence 

c'cs=  ec'v, 
ev  being  drawn  perpendicular  to  d'c'  produced.     Then 

c'v  =  c'e  '  cos  c'cs. 
Whence  cs   =  A  •  c'v. 

Now  c'v  =j  cos  <j>  —  p. 

Therefore  cs    =  A  (j  cos  <j>  —  jp),  (3) 

which  is  the  roll  of  the  wheel  for  the  motion  of  d  through 
the  arc  dd' '.     To  show  that   this   is  proportional  to  the  area 
dd'o'o,  there  must  be  deduced  an  expression  for  that  area. 
From  Trigonometry 


ed  =  V/2  +  h*  +  2jh  cos  <£, 
dd'=ed-A. 

The  area  of  a  sector  of  a  circle  is  the  product  of  one  half  its 
arc  by  its  radius,  whence 

Area  edd'     =  J  A  (/>  +  A2  +  2jh  cos  0).  (4) 

Using  the  value  of  the  radius  of  the  zero  circumference  given 
in  equation  (1),  there  results  for  the  value  of  the  area  eoo', 

Area  eoo'      =  1 A  (^  +  A2  +  2  pK).  (5) 

Subtracting  (5)  from  (4),  there  results 

Area  dd'o'o  =  Ah  (j  cos  <f>  -  p).  (6) 

R'M'D  SURV.  — 12 


178  LAND   SURVEY   COMPUTATIONS. 

This  area  is  equation  (3),  the  roll  of  the  wheel,  multiplied  by 
the  length  of  the  adjustable  arm,  arid  hence  is  proportional  to 
the  roll  of  the  wheel.  Q.  E.  D. 

It  is  now  to  be  shown  that,  in  tracing  a  closed  area,  the 
record  of  the  wheel  is  correctly  summed.  In  Fig.  81  let  the 
tracing  point  move  about  the  area  dd^d^d  clockwise.  Motion 
from  d  to  d^  will  cause  a  clockwise  roll  of  the  wheel  propor- 
tional to  the  area  dd^o'o.  The  motion  from  d1  to  d2  will  be  neu- 
tralized by  motion  from  ds  to  d.  Motion  from  d2  to  d3  will  cause 
counter-clockwise  roll  of  the  wheel  proportional  to  the  area 
c?2c?30o',  and  the  resulting  roll  will 
therefore  be  proportional  to  the 

area  dd-^d^dy 

The  student  may  reason  similarly 
x   for  the  other  areas.     Since  the  roll 
of   the  wheel  is  proportional  to  the 
area  lying  between  t?'s  path  and  the 
zero  circumference  and  is  positive,  or 
clockwise,  when  d  is  outside  the  zero 
circumference  and  moves  to  the  right, 
and  negative  when  d  is  inside  and 
FlG-  81g  moves  to  the  right,  it  follows  that  if 

an  area  is  traced  with  e  inside  that  area,  so  that  d  must  com- 
plete a  revolution  about  e,  there  must  be  added  to  the  area  ob- 
tained by  multiplying  the  roll  of  the  wheel  by  A,  the  area  of 
the  zero  circumference.  This  area  is 


159.  To  find  the  zero  circumference.     This  area  is  usually 
furnished  with  the  instrument  when  it  comes  from  the  maker, 
but  may  be  found  thus  : 

Measure  a  known  area  with  the  point  e  within  it  and  com- 
pare the  result  by  the  instrument  with  what  is  known  to  be 
the  correct  area.  The  difference  is  Z.  This  should  be  done 
a  number  of  times,  and  a  mean  value  of  the  several  determina- 
tions used. 

160.  To  find  the  circumference  of  the  wheel.      If  n  is  the 
number  of  revolutions,  and  c  is  the  circumference, 

Roll  of  wheel  =  no. 


THE   SLIDE   RULE. 


179 


In  a  given  circumscribed  area  with  e  outside, 
A  =  hnc. 

If  c  is  not  known,  measure  a  known  area  with  any  con- 
venient length  of  arm  and  note  the  reading  of  the  wheel,  which 
is  n.  From  the  known  quantities  compute  c.  This  should 
likewise  be  done  a  number  of  times. 

161.  Length  of  arm.  It  is  very  convenient  to  make  h  such 
a  length  as  will  reduce  the  work  of  multiplying  hnc  to  a  mini- 
mum. Most  instruments  are  so  made  that  the  length  of  the 
arm  may  be  such  that 

To  find  what  this  length  is  for  a  given  instrument  in  which 
c  is  known,  let  it  be  assumed  that  one  revolution  of  the  wheel 
shall  correspond  to  ten  square  inches  ; 
then  10  =  Ac, 

10 


and 


7, 

A  =  — -. 


If  the  arm  A  is  not  graduated,  it  may  be  set  by  trial  so  that 
A  =  10  n  and  the  value  c  will  not  be  required.  Some  cheap  forms 
of  the  instrument  are  made  with  the  arm  h  fixed  in  length. 
When  so  made  they  are  usually  proportioned  so  that  A  =  10  n. 

The  drawing  on  which  the  instrument  is  to  be  used  should 
be  perfectly  smooth 

THE  SLIDE  RULE.i 

162.  Described.  The  slide  rule  is  an  instrument  for 
mechanically  performing  multiplication,  division,  involution, 
and  evolution.  It  is  merely  a  series  of  scales,  which  are  the 
logarithms  of  numbers  laid  off  to 
scale,  so  arranged  that  by  sliding 
one  scale  on  the  other  the  logarithms 
may  be  mechanically  added  or  sub- 
tracted. The  divisions  are  numbered 
with  the  numbers  to  which  the  plotted 
logarithms  correspond. 

The  rule  is  constructed  in  many 
forms,  but  the  principles  involved  are  FIG.  82. 

1  Written  by  C.  W.  Crockett,  C.E.,  A.M.,  Professor  of  Mathematics  in  the 
Rensselaer  Polytechnic  Institute. 


180          LAND  SURVEY  COMPUTATIONS. 

the  same  in  all.  The  ordinary  rule,  about  ten  inches  long, 
consists  of  a  framework  called  the  rule  and  a  movable  part 
called  the  slide,  arranged  as  shown  in  Fig.  82.  On  their 
surfaces,  which  should  be  in  the  same  plane,  are  scales  at  I 
and  IV  on  the  rule,  and  at  II  and  III  on  the  slide.  The 
initial  points  of  these  scales  are  in  a  line  perpendicular  to  the 
upper  edge  of  the  rule.  A  runner,  acting  on  the  principle 
of  a  T-square,  assists  in  finding  points  on  the  scales  that  are 
at  a  common  distance  from  the  initial  points  of  the  scales. 

The  slide  may  be  inverted  —  turned  end  for  end  —  so  that 
II  is  adjacent  to  IV,  and  III  to  I,  or  reversed  so  that  the  other 
side  of  the  slide  becomes  visible.  One  form  of  the  slide  rule  is 
shown  in  Fig.  83. 


FIG.  83. 

163.  Historical.     In  1624  Gunter  proposed  the  use  of  the 
logarithmic  scale,  as  shown  in  Art.  165.     In  1630  Oughtred 
suggested  that  two  scales,  sliding  by  each  other,  could  be  used. 
In  1685  Partridge  fastened  two  scales  together  by  bits  of  brass, 
another  scale  sliding  between  them.     In  1851  Mannheim  intro- 
duced the  runner. 

164.  Construction  of  the  scales.     A  logarithmic  scale  is  one 
on  which  the  distance  from  the  initial  point  to  any  division  is 
proportional  to  the  mantissa  of  the  logarithm  of  the  number 
corresponding  to  that  division.     The  slide  rule  usually  bears 
two  of  these  scales,  constructed  as  follows  : 

B  I  T       ?     f"      -  ?    }  ?       9     f~      "?    1 


7  i I ?       i      ?     9    T    i  ?  i 

'a  6  c  d  e  f        g        h      i      3 

FIG.  84. 

On  scale  B,  Fig.  84,  let  a  distance  of  5  inches  represent  unity  in  the 
logarithm,  so  that,  if  a'j'  =  5",  we  have 


THE  SLIDE   RULE.  181 

log  1  =  0.00;  and  the  beginning  of  the  scale  is  marked  1. 

log  2  =  0.30;  and  at  b',  so  that  a'V  =  1.5",  we  mark  2. 

log  3  =  0.48,     "     '<   c',   "     "     a'c'  =  2.4",    "       «       3. 

log  4  =  0.60;     "     "  d',  "     "     a'<f  =  3.0",   "      "      4. 


log    8  =  0.90 ;  and  at  h',  so  that  a'h '  =  4.5",  we  mark  8. 

log  10  =  1.00;     "      "  /,  "      "     a'/ =  5.0",   "       "      1. 
This  is  duplicated  on  the  right  of/,  so  that  the  total  length  of  scale  B  is  10 
inches. 

On  scale  C,  let  a  distance  of  10  inches  represent  unity  in  the  logarithm, 
so  that,  if  aj  =  10",  we  have 

log    1  =  0.00 ;  and  the  beginning  of  the  scale  is  marked  1. 

log    2  =  0.30;  and  at  b,  so  that  ab  =  3.0",  we  mark  2. 

log    3  =  0.48;     "     "  c,  "     «     ac  =  4.8",    "       '<      3. 

log    4  =  0.60;     «     «  d,  "     "    ad  =  6.0",   "       "     4. 


log    8  =  0.90;  and  at  h,  so  that  ah  =  9.0",  we  mark  8. 
log    9  =  0.95;     «      «   i,  "      "      at  =  9.5",   "       "       9. 
log  10  =  1.00;     "      "  j,  "      "      rt/  =  10.0",«      "       1. 
The  initial  points,  a  and  a',  of  these  scales  are  in  the  same  vertical  line, 
and  aj  is  exactly  twice  the  length  of  a'j'. 

165.  Use  of  the  scales.  The  following  examples  will  show 
the  use  of  the  scales  : 

Suppose  we  wish  to  find  the  product  of  2  and  3,  using  scale  C.  The 
logarithm  of  the  product  is  equal  to  the  sum  of  the  logarithms  of  the  two 


FIG.  85. 

numbers.  Then,  if  with  a  pair  of  dividers  we  lay  off  rs  =  ab  =  log  2  and 
st  =  ac  —  log  3,  rt  will  represent  the  logarithm  of  the  product,  and  we  find  by 
comparison  with  scale  C  that  rt  =  n/=  log  6. 

To  divide  6  by  2  we  must  subtract  log  2  from  log  6.  Lay  off  with  the 
dividers  rt  =  af=  log  6  and  rs  =  ab  =  log  2.  Then  st  will  represent  the  log- 
arithm of  the  quotient,  and  by  comparison  with  scale  C  we  find  that 
st  =  be  =  log  3. 

166.  The  Mannheim  rule.  This  is  a  straight  rule,  in  which 
the  scales  I  and  II  are  constructed  in  the  way  described  for 
scale  B,  and  scales  III  and  IV  in  that  described  for  C. 

The  carpenter's  rule  is  also  a  straight  rule,  in  which  the 
scales  I,  II,  and  III  are  similar  to  B,  while  IV  is  similar  to  C. 


182 


LAND  SURVEY  COMPUTATIONS. 


The  scales  on  the   slide  will  be   denoted   by  A,  so   that   the 
arrangement  is  shown  in  Fig.  86. 

The  Thacher  rule,  described  in  Art.  182,  is  equivalent  to  a 
straight  carpenter's  rule  720  inches  long,  the  slide  bearing  only 
one  scale  A,  similar  to  B. 

This  explanation  of  the  slide  rule  will  be  confined  to  the 
carpenter's  rule  for  two  reasons  :  first,  the  principles  are  the 
same  for  all  the  rules,  and,  second,  the  explanation  will  also 
apply  to  the  Thacher  rule,  which  is  ex- 
tensively used  on  account  of  its  size  and 
consequent  accuracy. 

A  rule  may  be  easily  constructed 
that  will  be  of  great  assistance  in  fol- 
lowing the  explanations.  Cut  out  two 
pieces  of  stiff  paper  or  cardboard  about 
twelve  inches  long,  one  being  two  inches 
wide,  to  form  the  rule,  and  the  other 
one  half  inch  wide,  to  form  the  slide.  Along  the  middle  of  the 
wider  piece  draw  two  lines  one  half  inch  apart  and  lay  off  a 
scale  B  above  the  upper  line,  and  a  scale  O  below  the  lower. 
On  both  the  upper  and  the  lower  margins  of  the  narrower  piece 
lay  off  scales  similar  to  scale  B.  It  will  be  sufficient  to  lay 
off  the  distances  corresponding  to  the  whole  numbers  from  one 
to  ten,  the  logarithms  being  taken  from  any  table. 


167.    Use    of    the    rule. 

show  the  use  of  the  rule  : 


The   following   illustrations   will 


1.   Multiplication.     To  find  the  product  of  two  numbers,  say  2x3,  with 
the  rule,  set  1  of  scale  A  opposite  2  of  scale  B,  and  opposite  3  of  scale  A 


A       [ 

A 

2 

3        4 

FIG.  87. 

find  the  product  6  on  scale  B.     For  in  this  way  to  log  2  on  scale  B  we  add 
log  3,  found  on  scale  A,  giving  log  6  on  scale  B. 

If  2  is  to  be  multiplied  by  any  other  number,  as  4,  the  same  setting 
is  made,  and  opposite  4  of  A  we  find  the  product  on  B. 


THE   SLIDE   RULE.  183 

If  any  constant  number  b  is  to  be  multiplied  by  a  series  of  numbers,  so 
that  we  wish  to  find  the  value  of  bx  for  different  values  of  x  —  opposite  b  of 
B,  set  1  of  A  and  opposite  x  of  A  read  the  product  bx  on  B.  This  operation 
may  be  stated  as  follows,  reading  vertically,  the  lines  B  and  A  representing 
the  two  scales  used : 

B.      Opp.  b.         Read  bx. 

A.      Set  1.  Opp.  a;. 


2.    Division.     To  divide  a  number  by  another,  as  8  H-  2,  set  2  of  A  oppo- 
site 1  of  B,  and  opposite  8  of  A  read  the  quotient  4  on  B.     For  in  this  way 


1 
A. 


FIG.  88. 

we  take  away  log  2  from  log  8  on  scale  A  and  then  find  the  number  on  B 
that  corresponds  to  the  remainder. 

To  divide  6  by  2  we  could  use  the  same  setting,  reading  the  quotient  on 
B  opposite  6  of  A. 

To  divide  a  series  of  numbers  x  by  a  constant  number  a,  opposite  1  of 
B  set  a  of  A,  and  opposite  x  of  A  read  the  quotient  x  -f-  a  on  B. 


B.     Opp.  1.    Reads-:-  a. 
A.     Set  a.      Opp.  x. 

3.    Proportion.     An  expression  in  the  form  — ,  as  — - — ,  may  be  written 
-x  3.     Opposite  4  of  B  set  2  of  A,  and  the  index  (1)  of  A  will  be  opposite 


\*l 


4 M. 


FIG.  89. 

the  quotient  4  -s-  2  on  B ;  then  opposite  3  of  A  read  the  result  6  on  B,  for  in 
this  way  we  add  log  3  to  the  logarithm  of  the  quotient. 

If  a  and  b  are  constants  and  x  a  variable,  we  may  use  the  following 
setting  for  — ,  reading  vertically  : 


B.     Opp.  b.   Read  bx  H-  a. 
A.     Set  a.     Opp.  x. 


184  LAND   SURVEY    COMPUTATIONS. 

It  will  be  seen  that  —  becomes  bx  when  a  =  1,  and  -  when  &  =  *•     Com- 

a  a 

pare  the  settings  for  multiplication  and  division  with  that  for  proportion. 

168.    Infinite  extent  of  the  logarithmic  scale.     In  the  com- 

mon system  of  logarithms  a  single  mantissa  will  correspond  to 
a  given  sequence  of  figures,  the  position  of  the  decimal  point 
affecting  only  the  characteristic  ;  and  to  a  given  mantissa  a 
single  sequence  of  figures  will  correspond,  the  position  of  the 
decimal  point  being  fixed  by  the  characteristic.  Thus, 

log  3  =  0.48  ; 
log  30  =  1.48; 
log  300  =  2.48. 

If  we  make  aj  —  jk  —  Icl  =  nm  =  ma,  and  if  aj  =  log  10,  we  have, 

0.01  _  <U  _  1  _  10  _  100  _  1000 

n  m  a  j  k  I 

FIG.  90. 

ak  =  2  log  10  =  log  102  =  log  100; 
al  =  3  log  10  =  log  108  =  log  1000  ; 

am  =  -  log  10  =  log  10-1  =  log  ^-  =  log  0.1  ; 
an  =  -  2  log  10  =  log  10-2  =  log  _|_  =  log  0.01. 

Hence,  the  logarithm  of  a  number  between  10  and  100  will  be  repre- 
sented by  a  distance  greater  than  aj  and  less  than  ak.  The  logarithmic 
scale,  therefore,  is  not  merely  the  distance  aj,  but  it  is  composed  of  a  series 
of  such  parts,  extending  without  limit  on  both  sides  of  a.  Any  one  of  the 
points  n,  m,  a,  /,  k,  I,  and  so  on,  may  be  considered  as  the  starting  point 
(log  1  =  0);  thus,  if  the  scale  is  supposed  to  start  at  n,  we  would  mark  1  at 


FiG.  91. 

n,  10  at  m,  100  at  a,  1000  at  /,  and  so  on.     For  this  reason  the  indices  of 
scales  B  and  C  are  marked  1,  and  not  with  multiples  of  10. 

To  illustrate  this,  suppose  we  wish  to  multiply  5  by  8.  In  Fig.  85, 
using  scale  C,  make  rs  =  ae  =  log  5,  and  st  =  ah  =  log  8.  Then  comparing 
the  distance  rl  with  scale  C,  we  find  that  rt  is  greater  than  aj,  the  excess 
being  equal  to  ad  =  log  4.  Hence, 

log  (5x8)=  log  5  +  log  8  =  aj  +  ad  =  log  10  +  log  4  =  log  40. 

Again,  suppose  we  wish  to  divide  4  by  8.  In  Fig.  91,  make  st  =  log  4, 
and  tr  =  log  8.  Then,  sr  is  a  negative  mantissa,  and 

—  rs  =  —  ms  4-  mr  =  —  1  +  mr. 


THE   SLIDE   RULE.  185 

By  comparison  with  scale  C  we  find  that  mr  is  the  mantissa  of  log  5,  so 
that  the  quotient  is  0.5. 

169.  Sequence  of  figures.     In  operations  with  the  slide  rule, 
we  use  the  mantissas  of  the  logarithms  of  the  quantities  in- 
volved, and  the  distances  found  represent  the  mantissas  of  the 
results.     We  find,  therefore,  the   sequence   of   figures   in  the 
result,  the   position   of  the   decimal   point   being   determined 
either  by  special  rules  or  by  rough  computation. 

170.  Shifting  the  slide.     As  we  use  the  mantissas  only,  and 
as  the   addition  or  subtraction   of  the  distance   between   the 
indices  of  the  scales  would  affect  the  characteristic  alone,  we 
can  change  the  position  of  the  slide,  by  bringing  one  index  into 
the   position   previously  occupied   by  another   index,  without 
affecting  the  sequence  of  figures  in  the  result. 

171.  Use  of  the  runner. 

Let  us  find  the  value  of  2  x  3  x  4.  Opposite  2  of  B  set  1  of  A,  and  3 
of  A  will  be  opposite  6  (=  2  x  3)  on  B.  Instead  of  reading  this  number, 
set  the  runner  over  3  of  A.  Bring  1  of  A  to  the  runner,  and  now  read  the 
final  result  24  on  B  opposite  4  of  A . 

In  order  that  the  runner  may  be  used,  without  intermediate  reading,  it 
is  necessary  that  the  result  should  be  found  on  the  rule. 

172.  Squares  and  square  roots. 

The  distance  on  C  representing  log 2 (=3.0")  is  just  twice  that  repre- 
senting log  2  (  =  1.5")  on  B.  Hence,  if  x  is  any  number,  log  x  on  C  is  twice 
logx  on  B.  If  the  initial  points  of  the  two  scales  are  in  the  same  vertical 
line,  any  number  x  on  C  will  correspond  to  its  square,  x2,  on  B ;  for 

log  x  on  C  =  2  log  x  on  B  —  log  z2  on  B. 
Conversely,  any  number  on  B  will  correspond  to  its  square  root  on  C. 

1.  Squares.     To  find  the  square  of  any  number,  find  the  number  on  C, 
and  the  corresponding  number  on  B  will  be  its  square. 

2.  Square  roots.     To  find  the  square  root  of  any  number,  find  the  num- 
ber on  B,  and  the  corresponding  number  on  C  will  be  its  square  root.     To 
avoid  artificial  rules  it  is  necessary  to  know  approximately  the  first  figure  of 
the  root.     Divide  the  numbers  into  sections  of  two  digits  each,  commencing 
at  the  decimal  point,  as  in  arithmetic,  and  call  the  first  section  at  the  left 
that  contains  a  significant  figure  the  left  section.     If  this  left  section  is  10 
or  greater,  the  first  figure  of  the  root  will  be  3  or  greater,  and  if  it  is 
less  than   10,  the  first  figure  of  the  root  will  be  3  or  less.     In  the  car- 
penter's   rule    the  middle    index   of    B   corresponds  to  3.16+  on  C  since 


186 


LAND   SURVEY   COMPUTATIONS. 


VlO  =  3.1622 +  .  Therefore,  if  the  left  section  is  less  than  10,  the  number 
whose  square  root  is  required  is  found  on  the  left  scale  of  B,  and  if  greater 
than  10,  it  is  found  on  the  right  scale  of  jB.1 

173.  General  statement  of  problems.  If  a  and  b  are  con- 
stants, and  re  is  a  variable  number,  the  following  expressions 
may  be  solved,  each  with  one  setting,  with  the  carpenter's  or 
with  the  Thacher  rule,  the  slide  being  in  the  position  indicated 
and  the  result  being  read  on  the  scale  named : 

SLIDE  DIRECT. 
EXPRESSION.  READ  ON 


1.  ,=£ 

a 


Rule  (B). 
Rule  (C). 


3.  y  =  ~-  Rule  (£). 


Rule  (C). 


? 


Slide  (4). 
Slide  (A). 


SLIDE 

INVERTED. 

EXPRESSION. 

READ  ON 

8.  ,4° 

Rule  (B). 

9-  y=\^ 

-•         Rule  (C). 

10.    y  =  ^. 

Rule  (B). 

11.  y=\-x 

-•        Rule  (C). 

12.  y  =  -^ 

Slide  (A). 

13.  3,  =  ^. 

Slide  (A). 

7.   y  =  ^-  Slide  (A). 

In  these  expressions  either  a  or  b  may  be  equal  to  unity. 

174.   Slide  direct. 
1. 


bx 


.-.  log  y  =  (log  b  -  log  a)  4-  log  x* 

From  log  5  we  must  take  away  log  a,  and  then  add  log  x  to  the  differ- 
ence. Hence  log  x  must  be  found  on  the  same  scale  as  log  a,  the  result 
being  read  on  the  scale  on  which  log  b  was  found  (see  Fig.  89).  Reading 
vertically,  we  have 

B.      Opp.  b.          Read  y. 


A. 
A. 


Set  a. 


Opp.  x. 


C. 


1  Or  the  number  is  found  on  the  left  scale  of  B  when  there  is  an  odd  number  of 
digits  to  the  left  or  an  odd  number  of  ciphers  to  the  right  of  the  decimal  point; 
otherwise  it  is  found  on  the  right  scale  of  B. 

*  The  quantity  in  the  parentheses  will,  in  every  case,  indicate  the  setting. 


THE   SLIDE   RULE.  187 

Ex.   6  =  3,   0  =  2.     a:  =  4,   y=    6; 
x  =  6,   y=    9; 

1  =  8,      =  12. 


Use  the  same  setting  as  in  the  preceding  case,  but  read  the  result  on  C 
instead  of  on  B  (see  Art.  172). 


B.      Opp.  b. 


A .      Set  a.  Opp-  x- 

A. 

C.  Read  y. 

Note  that  the  square  roots  are  always  read  on  C. 

When  a  number  on  C  is  to  be  compared  with  a  number  on  A,  it  will  be 
more  convenient,  in  the  carpenter's  rule,  to  use  the  scale  A  that  is  adjacent 
to  C.  Thus,  in  this  case  we  would  find  x  on  the  lower  scale  of  the  slide. 
With  this  caution,  the  upper  scale  of  the  slide  will  be  used  in  all  the 
settings. 


.- .  log  y  =  (log  b2  —  log  a)  +  log  x. 

Since  b  is  squared,  we  find  b  on  C,  so  that  the  number  opposite  to  it  on 
Bis  P. 


B.  Read  y. 

A.     Seta.  Opp.  a;. 

A. 

C.  Opp.  b. 

Ex.    b  =  3,   a  =  2 ;     x  =  4,   y  =  18 ; 


4.   j,=<V— • 

.  • .  log  y  =  \  {(log  52  -  log  a)  +  log  x}. 


1',. 


A.     Set  a.  Opp-  x. 

A. 

C.     Opp.  6.          Read  y. 


188          LAND  SURVEY  COMPUTATIONS. 


.-.  log y  =  (log  a  -  log  62)  +  log  x. 


B.  Opp.  z. 

A.     Seta.  Ready. 

A. 

C.  Opp.  b. 

In  this  case  we  must  read  on  the  slide ;  for  i2  and  x  must  both  be  found 
on  the  rule  or  both  on  the  slide,  the  latter  case  being  impossible,  since  the 
slide  does  not  contain  a  scale  similar  to  C. 

Ex.  a  =  6,  6  =  3;  x  =  2,  y  =  1.33; 
x  =  4,  y  =  2.67; 
x  =  S,  y  =  5.33. 

Note  that  Vy  can  not  be  determined  without  first  reading  the  value  of 
y,  since  the  slide  does  not  contain  a  scale  corresponding  to  C. 


6  ' 

.  • .  log  y  =  (log  a  —  log  6)  +  log  x2. 


A .     Set  a.  Read  y. 

A. 

C.  Opp.  x. 

Ex.  a  =  3,  6  =  2;  x  =  4,  y  =  24; 
x  =  6,  y  =  54; 
x  =  8,  y  =  96. 


.  • .  log  y  =  (log  a  -  log  62)  +  log  x2. 


A.      Seta.  Ready. 

C.      Opp.  6.  Opp.  x. 

Ex.   a  =  3,   6  =  2;     x  =  3,  y  =    6.75; 
x  =  6,   v  =  27.00; 


THE   SLIDE   RULE.  189 

175.  Slide  inverted.  The  arithmetical  complement  of  the 
logarithm  (written  colog)  of  a  number  is  the  logarithm  of  its 
reciprocal.  The  mantissa  of  the  cologarithm  of  a  number  is 
found  by  subtracting  the  mantissa  of  the  logarithm  from  unity. 
Thus 

mantissa  of  log  2       =    .301030 

mantissa  of  log  0.5    =    .698970 

.-.  sum  of  mantissas  =  1.000000 

Hence,  if  in  Fig.  84  ab  represents  log  2  on  C,  bj  will  represent  colog  2,  since 
bj  =  aj  -ab  =  log  10  -  log  2  =  1  -  log  2. 

With  the  slide  inverted.  Lhe  numbers  on  the  slide  increase  from  right  to 
left,  so  that  the  distance  from  the  left  index  to  a  number  represents  the 
mantissa  of  the  cologarithm  of  that  number. 

,  ,.* 

.-.  log  y  =  log  b  +  log  a  -  log  x 

=  (log  b  -  colog  a)  +  colog  x. 

Opposite  b  of  B  set  a  of  A,  thus  subtracting  colog  a  from  log  b  and 
placing  the  index  (1)  of  A  opposite  the  product  ba  on  B.  Then  opposite 
x  of  A  read  y  on  B. 


B.      Opp.  b.          Read  y. 
Inv.i  j  A.      Seta.  Opp.  x. 


Ex.  a  =  5,  6  =  12;  x  =  2,  #  =  30.0; 
x  =  4,  y  =  15.0; 
z  =  8,  y=  7.5. 


.• .  log  y  =  \  {(log  b  —  colog  a)  +  colog  x}. 


B.      Opp.  b. 


Inv.-H'      Sefca>  Opp.  x. 

(  A. 

C.  Read  y. 


i  Inverted. 


190          LAND  SURVEY  COMPUTATIONS. 

10.  y=*^. 
x 

.'.  log  y  —  (log  ft2  —  colog  a)  +  colog  x. 


B.  Read  y. 

Iny  (  A.      Seta.  Opp.  x. 

\  A . 

~C.      Opp.  h. 

Ex.  a  =  15,  b  =  2  ;  x  =  3,  y  =  20.0 ; 
x  =  5,  y  =  12.0 ; 
z  =  8,  y=  7.5; 


.'.  logy  =  I  {(log  fc2  -  colog  a)  +  colog  a;}. 


B. 


j        (  A .      Set  a.  Opp-  x- 

I  A. 

C.      Opp.  b.          Read  y. 

In  the  other  two  cases  the  result  must  be  read  on  the  slide,  so  that  the 
distances- from  the  left  index  (1)  will  be  the  mantissas  of  the  cologarithms  of 
the  corresponding  numbers,  while  the  distances  on  the  rule  will  correspond 
to  the  mantissas  of  the  logarithms. 

12    ,  =  *«. 
a:2 

.'.  colog  y  =  colog  a  +  colog  b  —  colog  z 
=  (colog  a  —  log  b)  +  log  x2. 


B.         Opp.  b. 


Inv(/l.        Seta.         Ready. 
~CT~  Opp.  x. 


Ex.  a  =  5,  6  =  12;  x=2,  y  =  15.00; 
x  =  4,  #=  3.75; 
a;  =  5,  y=  2.4. 


13.   y  =  —  • 
x2 


.'.  colog  y  =  colog  a  +  colog  62  —  colo£r  x2 
=  (colog  a  -  log  62)  +  log  x2. 


THE    SLIDE   RULE. 


I  A. 

C.       Opp.  b.      Opp.  x. 

Ex.  a  =  15,   6  =  2 ;     x  =  3,  y  =  6.67 ; 

x  =  5,'  y  =  2.4.  ' 

NOTE.  —  Expressions  similar  to  (1),  (8),  and  (10),  Art.  173,  may  be 
solved  iu  such  a  way  that  the  result  may  be  read  on  the  slide  A. 

176.    Use  of  the  runner  in  complicated  expressions. 

If  we  have  found  the  value  of  an  expression,  say  — ,  and  marked  its 
place  on  the  rule  by  placing  the  runner  there,  then  to  multiply  it  by  a  frac- 
tion -  we  bring  the  denominator  e  on  the  slide  to  the  runner  and  read  the 
result  on  the  rule  opposite  d  on  the  slide.  For  this  subtracts  log  e  from 
log  —  and  then  adds  log  d  to  the  remainder. 

_  abcde  _  ab     c    d     e 
ghk        g      h     k     1 

Denoting  the  runner  by  R,  we  have 

B.     Opp.  a.  Read  y. 

A.     Set  g.        R  to  b.     h  to  R.     R  to  c.     k  to  R.     R  to  d.     1  to  R.     Opp.  e. 

EXAMPLES. 

4  •  5  •  6  .  7_4  -  5     6    7=g5 
2-3-4     "~    2     '§'1     ' 

5-6.7-8      5-6781 


2-4 • 5- 6        2       4 
8-9  8-91 


=  0.5. 


2-3-4-6        2       34 
An  expression  of  the  form 

_a(m2  +  n2  +  r  +  s/>  +  93) 
y~  b 

may  be  written 

6  b          b         b          b 

and  the  values  of  the  several  terms  found  separately.    Notice  that  one  setting 
will  answer  for  the  first  three  terms,  if  the  results  are  read  on  the  slide. 


192  LAND   SURVEY  COMPUTATIONS. 

177.  Gage  points.  When  a  formula  is  often  used,  as  that 
for  computing  the  horse  power  of  an  engine,  it  will  be  found 
convenient  to  combine  the  constant  factors,  sometimes  using 
the  resulting  constant,  and  sometimes  its  reciprocal.  Such  a 
constant  is  called  a  gage  point. 

Thus  the  formula  for  the  horse  power  of  an  engine  is 

UP  _  P  x  0.7854  d2  x  2s  x  r 
33000 

where  p  =  mean  effective  steam  pressure  in  the  cylinder  in-lb.  per  sq.  in., 
d  =  diameter  of  piston  in  inches  ; 
s  =  stroke  in  feet, 
r  =  number  of  revolutions  per  minute. 

0-7854  x  2  _      1  d>r 

33000      "  21008  '  ~  21008' 

where  21008  is  the  gage  point  for  the  formula 


B.  Read  H.P. 

A.       Set  21008.         R  to  p.         1  to  R.        R  to  s.         1  to  R.        Opp.  r. 
A. 

C.  Opp.  d. 


178.   Extraction  of  cube  roots. 

y  =  j/b.  Since  ^8. 000  =  2,  ^80.000  =  4  +,  ^800.000  =  9  +,  the  cube 
root  of  any  sequence  of  figures  will  depend  upon  the  position  of  the  decimal 
point,  so  that  the  first  figure  of  the  root  should  be  known  approximately  in 
order  that  the  wrong  number  may  not  be  taken.  This  first  figure  may  be 
found  by  dividing  the  given  number  into  sections  containing  three  digits 
each,  commencing  at  the  decimal  point,  and  extracting  the  cube  root  of  the 
left  section.  Then,  with  the  slide  inverted,  opposite  b  of  B,  set  one  of  the 
extreme  indices  (1)  of  A,  and  find  the  number  on  C  that  is  opposite  the  same 
number  on  A.  This  will  be  the  cube  root  required.  Sometimes  the  right 
index  of  A  must  be  used,  at  others  the  left  index. 


Opp. 


'     SetL  °PP-y- 

A.  or  Opp.  y. 

C.  Read  y 


THE   SLIDE   RULE.  193 

179.  Slide  reversed.     The  reverse  of   the   slide  sometimes 
has  along  one  edge  a  scale  of  logarithmic  sines  and  along  the 
other  a  scale  of  logarithmic  tangents,  the  scales  being  so  ar- 
ranged that  the  numbers  (degrees)  increase  from  left  to  right 
when,  the  slide  being  reversed,  the  corresponding  edge  of  the 
slide  is  adjacent  to  B. 

180.  Scale  of  logarithmic  sines,  S. 

If  we  reverse  the  slide  and  place  S  adjacent  to  B  so  that  their  initial 
points  coincide,  we  find  that  the  beginning  of  B  corresponds  to  0°  34'  +  on 
S,  the  middle  index  of  B  to  5°  44'  +,  and  the  right  index  of  B  to  90°;  for 

log  sin  0°  34'  +  =  8.00  (  =  -  2), 
log  sin  5°  44'  +  =  9.00  (=-1), 
log  sin  90°  =  0.00, 

unity  in  the  characteristic  being  represented  by  the  same  distance  on  S  and 
on  B. 

In  this  position  the  natural  sine  of  an  angle  may  be  found  by  reading 
the  number  on  B  corresponding  to  the  angle  on  S. 

The  distance  from  the  beginning  of  5  to  any  division,  in  the  direction 
of  increasing  numbers,  represents  the  mantissa,  or  the  mantissa  +  1,  of  the 
logarithmic  sine  of  the  angle  corresponding  to  that  division ;  and  the  dis- 
tance from  any  division  to  the  other  end  (90°)  of  S  represents  the  mantissa, 
or  the  mantissa  +  1,  of  the  cologarithm  of  the  sine. 

In  expressions  containing  cos  x,  since  cos  x  =  sin  (90°  —  x),  we  may  sub- 
tract the  angle  x  from  90°  and  then  use  the  sine  of  the  remainder,  finding 
it  on  S. 

1.  a  =  c  sin  x. 

.-.  logo  =  logc  +  log  sin  x  =  logc  -  colog  sins. 

Opposite  c  of  B  set  the  beginning  of  S,  and  opposite  x  of  S  read  a  on 
B,  for  this  adds  log  sin  x  to  log  c.  If  x  should  fall  outside  of  the  rule,  we 
would  set  the  right  end  of  S  opposite  c  of  B,  and  then  opposite  x  of  S  read 
a  on  B,  for  this  subtracts  colog  sin  x  from  log  c. 

2.  6  =  ccosx. 

.• .  log  b  =  log  c  +  log  sin  (90°  -  a:)  =  log  c  -  colog  sin  (90°  -  x), 

and  the  methods  given  for  the  multiplication  of  a  sine  by  a  number 
(a  =  c  sin  x)  are  applicable,  using  90°  —  x  instead  of  x. 

3.  c  =  a  -4-  sin  x. 

.  • .  log  c  =  log  a  —  log  sin  x  =  log  a  +  colog  sin  x. 

Opposite  a  of  B  set  x  of  S,  and  opposite  the  end  of  S  read  c  on  B  ;  for 
this  subtracts  log  sin  x  from  log  a  when  we  read  opposite  the  left  end,  and 
adds  colog  sin  x  to  log  a  when  we  read  opposite  the  right  end. 
R'M'D  SURV.  — 13 


194  LAND   SURVEY  COMPUTATIONS. 

4.  c  =  b  -4-  cos  x. 

:.  log  c  =  log  b  -  log  sin  (90°  -  a;)  =  log  b  +  colog  sin  (90°  -  x), 
and  the  methods  for  c  =  a  •*-  sin  a;  are  applicable,  using  90°  —  z  instead  of  x. 

5.  a  —  c  sin  z  -4-  sin  z. 

.'.  log  a  =  log  c  —  log  sin  z  +  log  sin  x. 

Opposite  c  of  B  set  2  of  5,  and  opposite  x  of  5  read  a  on  .6 ;  for  this 
first  subtracts  log  sin  z  from  log  c,  and  then  adds  log  sin  x  to  the  remainder. 
If  x  falls  outside  the  rule,  set  the  runner  over  the  end  of  S  that  is  in  the 
rule,  shift  the  slide  until  the  other  end  of  S  comes  to  the  runner,  and  then 
opposite  x  of  S  read  a  on  B.1 

The  expression  for  a  becomes  c  sin  x  when  z  =  90°,  and  c  -f-  sin  z  when 
x  =  90°.  Compare  the  setting  just  given  with  those  for  c  sin  x  and  a  -f-  sin  x. 

6.  sin  x  =  a  sin  z  -T-  c. 

.'.  log  sin  x  =  log  sin  z  —  log  c  +  log  a. 

Opposite  c  of  .B  set  z  of  5,  and  opposite  a  oi  B  read  a;  on  S ;  for  this 
subtracts  log  c  from  log  sin  z,  and  adds  log  a  to  the  remainder. 
This  becomes  sin  x  =  a  -t-  c  when  z  =  90°. 

7.  cos  a:  =  6  -i-  c. 

.'.  log  sin  (90°  -  z)  =  log  b  -  log  c 

=  log  sin  90°  -  log  c  +  log  b. 
Opposite  c  of  B  set  90°  of  S,  and  opposite  b  of  £  read  90°  -  x  on  5. 

181.   Scale  of  logarithmic  tangents,  T. 

If  we  reverse  the  slide  and  place  T  adjacent  to  B  so  that  their  initial 
points  coincide,  we  find  that  the  beginning  of  -B  corresponds  to  0°  34'  +  on 
T,  the  middle  index  of  B  to  5°  42'  +  ,  and  the  right  index  to  45° ;  for 

log  tan  0°  34' +  =  8.00  (=-2), 
log  tan  5°  42'+  =  9.00  (=-1), 
log  tan  45°  =  0.00. 

Unity  in  the  characteristic,  therefore,  corresponds  to  the  same  distance  on 
T,  A,  and  S. 

In  this  position  the  natural  tangent  of  an  angle  less  than  45°  may  be 
found  by  reading  the  number  on  B  opposite  the  given  angle  on  T.  To  find 
the  tangent  of  an  angle  greater  than  45°,  invert  the  slide  so  that  T  is  adja- 
cent to  C,  their  ends  coinciding,  and  read  the  number  on  B  corresponding 
to  the  complement  of  the  angle  on  T. 

The  distance  from  the  beginning  of  T  to  any  division,  in  the  direction 
of  increasing  numbers,  represents  the  mantissa,  or  the  mantissa  +  1,  of  the 
logarithmic  tangent  of  the  angle  corresponding  to  that  division ;  while  the 

1  In  all  cases  when  the  result  cannot  be  read,  the  slide  is  shifted  over  the  length 
of  its  scales,  since  this  merely  changes  the  characteristic. 


THE   SLIDE   RULE.  195 

distance  from  that  division  to  the  end  (45°)  of  the  scale  represents  the  man- 
tissa, or  the  mantissa  +  1,  of  the  cologarithm  of  the  tangent 
Since  tan  x  =  1  -f-  cot  z.  we  have 

log  tan  x  —  colog  cot  x 
and  colog  tan  x  =     log  cot  x. 

Hence  the  distance  that  represents  log  tan  x  also  represents  colog  cot  x,  and 
that  representing  colog  tan  x  also  represents  log  cot  x. 

When  x  is  greater  than  45°,  subtract  x  from  90°  and  use  the  cotangent 
of  the  remainder  instead  of  tan  x,  and  the  tangent  of  the  remainder  instead 
of  cot  a;. 

1.  a  =  b  tan  x. 

x  <  45°.     .'.  log  a  =  log  b  +  log  tan  x  =  log  b  —  colog  tan  x. 

Opposite  b  of  B  set  the  end  of  T,  and  opposite  x  of  T  read  a  on  B ;  for 
this  adds  log  tan  x  to  log  b  when  the  left  end  is  used,  and  subtracts  colog  tan 
x  from  log  b  when  the  right  end  is  used. 

x  >  45°.  .'.  log  a  =  log  b  +  log  cot  (90°  -  x) 
=  log  b  -  log  tan  (90°  -  x) 
=  log  b  +  colog  tan  (90°  -  z). 

Opposite  b  on  B  set  90°  -  x  of  T,  and  opposite  the  end  of  T  read  a  on 
B;  for  this  subtracts  log  tan  (90°  —  z)  from  log  b  when  the  left  end  is  used, 
and  adds  colog  tan  (90°  -  x)  to  log  b  when  the  right  end  is  used. 

2.  b  =  a  -f-  tan  x. 

x  <  45°.     .'.  log  b  =  log  a  —  log  tan  x  =  log  a  +  colog  tan  x. 

Opposite  a  of  B  set  x  of  T  and  opposite  the  end  of  T  read  b  on  B ;  for 
this  subtracts  log  tan  x  from,  or  adds  colog  tan  x  to,  log  a  when  we  read 
opposite  the  left  or  the  right  end  respectively. 

x  >  45°.  /.  log  b  =  log  a  -  log  cot  (90°  -  x) 
=  log  a  +  log  tan  (90°  -  x) 
=  log  a  -  colog  tan  (90°  -  x). 

Opposite  a  of  B  set  the  end  of  T  and  opposite  90° -x  of  T  read  b  on  J5; 
for  this  adds  log  tan  (90°  -  x)  to,  or  subtracts  colog  tan  (90°  -  x)  from, 
log  a  when  we  use  the  left  or  the  right  end  respectively. 

3.  b  =  a  cot  x. 

/.  6  =  a  -=-  tan  x, 

and  the  settings  given  under  (2)  are  used. 

4.  b  =  a  +-  cot  x. 

:.b  =  a  tan  x, 

and  the  settings  given  under  (1)  are  used. 


196 


LAND   SURVEY   COMPUTATIONS. 


182.  The  Thacher  rule.  This  rule  (Fig.  92)  consists  of  a 
woovlen  base  bearing  two  upright  metallic  standards  with  a 
large  circular  opening  in  each,  the  line  joining  the  centers  of 
the  openings  being  perpendicular  to  the  planes  of  the  standards. 
Attached  to  the  standards  are  two  circular  plates  of  metal, 


FIG.  92. 

each  with  a  large  circular  opening  concentric  with  those  in  the 
standards,  so  arranged  that  they  can  revolve  around  the  line 
joining  the  centers  of  the  openings  as  an  axis.  These  plates 
are  united  by  twenty  bars  a  little  more  than  18  inches 
long,  triangular  in  section,  and  perpendicular  to  the  plates. 
These  bars  are  arranged  at  equal  dis- 
tances around  the  circular  openings, 
with  their  vertices  outward  so  that  their 
bases  form  a  cylindrical  envelope,  the 
distances  between  the  bars  being  ap- 
proximately equal  to  the  width  of  their 
bases.  A  cylindrical  slide  fits  in  this 
cylindrical  envelope,  moving  with  either 
a  rotary  or  a  longitudinal  motion.  The 

system  of  bars  and  plates  may  be  rotated  about  the  axis  of  the 
cylindrical  envelope  without  disturbing  the  relative  position  of 
the  bars  and  the  slide. 

The  system  of  bars  is  the  rule,  and  the  cylinder  is  the  slide. 
The  latter  bears  only  one  scale  A,  while  the  former  contains 
the  scales  B  and  C.  The  Thacher  rule  is  equivalent  to  a 
carpenter's  rule  in  which  the  slide  bears  only  one  of  the 
two  A  scales.  The  arrangement  of  the  scales  is  practically 
the  same  as  that  in  a  straight  rule  of  the  form  shown  in 
Fig.  93. 

The  scale  A  is  laid  down  on  the  cylinder  as  follows : 


THE  SLIDE   RULE.  197 

Let  M  and  N  be  two  logarithmic  scales  360  inches  long,  and  let  each  be 
divided  into  forty  equal  parts,  a,  b,  •••,  a',  b',  •••.  Draw  forty  equidistant 
elements  on  a  cylinder  and  lay  off  on  them  the  segments  of  the  scales  as 
shown  in  Fig.  95,  which  represents  the  development  of  the  cylinder.  Then 
the  scales  will  read  continuously  from  left  to  right. 

The  twenty  bars  are  graduated  on  each  side  and  bear  two  different 


FIG.  94. 


scales.  The  one  nearer  the  cylindrical  slide  B  is  constructed  in  the  same 
way  as  that  on  the  slide.  The  outer  one  C  is  formed  by  dividing  a  loga- 
rithmic scale  720  inches  long  into  eighty  equal  parts  and  placing  them  in 
order  above  a,  6,  -,  k,  1;  a',  b',  ...,  k',  V. 

The  Thacher  rule  gives  results  that  should  never  be  in 
error  by  more  than  one  unit  in  the  fourth  significant  figure, 
while  the  fifth  figure  can  often  be  found  with  only  a  small 
error.  The  error  should  not  exceed  one 
part  in  ten  thousand,  so  that  this  rule  is 
intermediate  in  accuracy  between  a  four- 
and  a  five-place  logarithmic  table. 


183.  Settings  for  the   Thacher   rule. 

The  settings  given  for  the  carpenter's  rule 
will  answer  for  the  Thacher  rule.1  All 
the  cases  that  can  be  solved  with  a  single 
setting  of  the  instrument  have  been  men- 
tioned. Problems  that  can  not  be  solved 
with  one  setting  may  sometimes  be  easily 
computed  by  the  use  of  the  runner,  and 
sometimes  by  computing  the  different  *1Q*  ^ 

parts  of  the  expression,  reading  the  results,  and  then  combining 
them  with  the  aid  of  the  rule. 

184.  Fuller's  slide  rule.     This  rule  is  shown  in  Fig.  96. 
The  hollow  sleeve  (7,  which  bears  the  graduations,  is  capable 
of  sliding  along  and  revolving  about  the  continuous  cylinder 
HH,  the  latter  being  held  by  the  handle  attached  to  it.     F  is 

1  Mr.  Thacher  has  devised  two  simple  statements  that  give  the  settings  for  all 
the  expressions  in  Art.  173  with  the  exception  of  the  third. 


198  LAND   SURVEY   COMPUTATIONS. 

a  fixed  index  fastened  to  H,  and  by  moving  the  sleeve,  any 
division  of  the  scale  may  be  made  to  coincide  with  F.  The 
cylinder  H  is  hollow,  forming  a  guide  for  the  motion  of  a  third 
cylinder  that  is  attached  to  the  flange  6r,  its  axis  being  coinci- 
dent with  the  common  axis  of  0  and  H.  A  and  B  are  two 
indices  fixed  to  and  moving  with  6r,  the  distance  AB  being 


FIG.  96. 

equal  to  the  axial  length  of  the  scale.  This  scale,  500  inches  in 
length,  is  wrapped  on  the  sleeve  O  in  the  form  of  a  helix,  its 
beginning  being  at  the  end  towards  6r.  Stops  are  provided, 
so  that  the  indices  may  be  readily  made  to  coincide  with  the 
beginning  of  the  scale. 

To  illustrate  the  use  of  the  indices,1  suppose  that  we  wish  to  find  the 
value  of  a  x  ft.  Move  the  sleeve  C  until  a  is  opposite  F,  and  then  by  mov- 
ing G,  place  A  at  the  beginning  of  the  scale ;  the  distance  from  A  to  F, 
along  the  helix,  will  be  log  a.  Move  the  sleeve  C,  being  careful  not  to 
change  the  relative  positions  of  G  and  H,  until  b  is  opposite  A  ;  then  the 
distance  along  the  helix  from  the  beginning  of  the  scale  to  A  is  log  b. 
Hence  the  distance  from  the  beginning  of  the  scale  to  F,  along  the  helix,  is 
log  b  +  log  a  =  log  ab,  so  that  the  product  will  be  found  on  the  scale  op- 
posite jP. 

185.  The  Mannheim  rule.  With  the  Mannheim  rule  (Art. 
166)  we  can  find  with  one  setting  the  value  of  any  expression 
in  a  fractional  form  with  two  factors  in  the  numerator  and  one 
in  the  denominator,  one  of  the  numbers  being  variable  and  one 
or  all  of  the  numbers  being  squared,  and  also  the  value  of  the 
square  root  of  such  an  expression,  the  result  being  always  read 
on  the  rule. 

The  carpenter's  and  the  Thacher  rules  do  not  possess  this 
power,  since  the  slide  does  not  bear  a  scale  similar  to  0. 

1  This  is  intended  only  to  illustrate  the  principle  of  the  instrument. 


BOOK   II. 

GENERAL   SURVEYING   METHODS. 


CHAPTER  VII. 


LAND  SURVEYS. 

186.  Obstacles  and  problems.  In  the  field  work  of  the  sur- 
veyor, various  obstacles  arise  which  must  be  overcome.  A  few 
of  the  more  common  difficulties,  with  the  methods  of  surmount- 
ing them,  will  be  given. 

I.  In  Fig.  97  it  is  required  to  produce  the  line  XA  in 
distance  and  direction. 

(1)  At  A  erect  the  perpendicular  AC,  to  which  erect  the 
perpendicular  CD,  which  make  long  enough  to  pass  the  obstacle. 
At    the    point    D 

erect  the  perpen-  r- 
dicular  DB  equal 
to  AC,  and  at  B 
erect  the  perpen- 
dicular BY,  which 
is  the  line  pro- 
duced. The  dis- 
tance A  B  equals 
CD. 

(2)  At   A   set 
off  an  angle  of  60° 
and    measure    A  C 
of  such  length  that 
a  line  making   an 

angle  of  60°  with  AC  will  pass  the  obstacle.  At  C  set  off 
from  A  C  an  angle  of  60°  and  measure  CB  equal  to  A  C.  At 
B  set  off  the  angle  ABC  60°.  The  instrument  will  then  be  in 
the  line  XA  produced,  pointing  toward  A,  and  the  distance 
AB  equals  A  C  or  CB. 

(3)  At  A  set  off  any  convenient  angle  BA  C  and  measure 
any  distance   AC.     At   C  set  off  any  angle  ACB,  making  it 


FIG.  97. 


202 


LAND   SURVEYS. 


90°  if  convenient,  and  measure  CB,  a  distance  determined  by 
solving  the  triangle  ABC,  and  at  B  set  off  an  angle  ABO  de- 
termined in  the  same  way. 


FIG.  98. 

II.  In  Fig.  98  it  is  required  to  determine  the  distance  AB, 
B  being  an  inaccessible  point,  or  the  stream  too  wide  to  be 
spanned  by  a  single  chain  or  tape  length. 

(1)  At  A  set  off  the  angle  BA  C  equal  to  90°  and  measure 
A  O  a  convenient  distance,  not  so  short  as  to  make  the  angle 


FIG.  99. 

ABC  less   than  20°,   unless  this   is  unavoidable,    and   at    C 
measure  the  angle  BOA.     Solve  the  right-angled  triangle. 

(2)    At  A  set  off  any  convenient  angle  BA  C  and  measure 
any  convenient  distance  AC,  observing  the  previous   caution 


TWO   COMMON   PROBLEMS. 


203 


with  regard  to  the  angle  at  B.     Measure  the  angle  at  C  and 
solve  the  triangle  ACB. 

III.  In  Fig.  99,  it  is  required  to  determine  the  length  and 
bearing  of  the  inaccessible  line  AB. 

(1)  Measure  the  line  CD,  from  each  end  of  which  both  A 
and  B  may  be  seen,  and  determine  its  bearing.  With  the 
instrument  first  at  O  and  then  at  D  measure  three  angles  at 
each  point.  Solve  the  triangle  A  CD  for  AC,  BCD  for  CB, 
and  ABC  for  AB,  and  an  angle  at  A  and  B.  Another  com- 
bination of  triangles  could  be  used. 

IV.  In  Fig.  100,  it  is  required  to  determine  the  length  and 
bearing  of  the  line  AB. 


FIG.  100. 

(1)  Run  the  random  line  AbcdB,  noting  bearings  and 
distances.  If  the  survey  is  treated  as  a  closed  field,  AB  will 
be  the  error  of  closure,  or  the  closing  line,  or  a  wanting  side 
which  may  be  fully  determined  in  the  manner  previously  given 
for  finding  the  bearing  and  length  of  a  wanting  side. 

187.  Two  common  problems.  I.  In  Fig.  101,  it  is  required 
to  determine  the  lengths  and  bearings  of  the  sides  of  the  field 
ABCDEF,  the  lines  being  through  woods  and  no  two  corners 
visible,  the  one  from  the  other.  The  positions  of  the  corners 
are,  however,  known. 

Run  the  random  line  AabBcdCefDgEhikFIA,  and  com- 
pute the  latitudes  and  longitudes,  or  the  coordinates,  of  the 
points  ABCDEF,  referred  to  the  true  or  any  convenient 
meridian,  preferably  the  true  meridian  through  the  most 
westerly  point  of  the  survey.  From  these  coordinates  deter- 


204  LAND  SURVEYS. 

mine  the  latitude  and  longitude  differences  and  the  bearing 
and  length  of  each  course. 

II.  If  a  corporation,  as  a  mining  company,  proposes  to 
purchase  a  considerable  tract  of  ground  containing  many  small 
parcels  owned  by  different  individuals,  and  a  description  of 
each  parcel  with  its  area  is  wanted,  there  are  two  ways  of 
obtaining  the  information. 


(1)  Each  parcel  may  be  separately  surveyed,  determining 
the  bearings  and  lengths  of  the  courses  and  the  areas. 

(2)  By  far  the  quicker  and  better  way  is  to  run  a  random 
field,  touching  on  the  various  corners  of  the  various  parcels, 
to  compute  the  coordinates  of   these  corners,  and  from  these 
the  bearings,  or  azimuths,  and  the  length  of  the  sides  and  the 
areas.     If  a  separate  description  of  each  piece  is  not  required, 
but  only  the  area,  the  computation  of  bearing  and  length  of 
course  may  be  omitted  and  the  areas  may  be  obtained  at  once 
from  the  coordinates  of  the  corners. 

SURVEYING  WITH  THE  CHAIN  ALONE. 

188.  In  most  surveys  the  chain  or  tape  is  used  in  connec- 
tion with  some  instrument  for  measuring  angles,  since,  when 
the  sides  and  angles  of  a  polygon  (a  field)  are  known,  the 
polygon  may  be  drawn  and  its  area  computed.  But  it  is 
frequently  convenient  in  approximating  to  make  an  entire  sur- 
vey, usually  of  a  very  small  tract,  with  the  chain  alone.  This 
entire  survey  may  be  a  part  of  a  larger  survey,  but  is,  never- 


SURVEYING  WITH  THE  CHAIN,  205 

theless,  being  a  closed  survey,  complete  in  itself.  A  method  of 
making  such  a  survey  will  therefore  be  described  briefly,  intro- 
ducing some  methods  that  are  employed  as  well  when  the  com- 
pass or  transit  is  used  with  the  chain. 

189.  Preliminary  examination.  Let  Fig.  102  represent  the 
map  of  a  farm,  a  survey  of  which  is  desired,  and  let  it  be  sup- 
posed that  there  is  no  instrument  available  except  a  chain  or 
tape.  It  will  of  course  be  impossible  to  determine  bearings. 
It  is  assumed  that  it  is  the  area  that  is  desired. 

The  first  thing  to  do  in  any  land  survey  is  to  make  a  rough 
sketch  of  the  tract  to  be  surveyed,  drawing  it  as  nearly  as  pos- 
sible in  correct  proportion,  from  an  inspection  made  by  walking 
over  the  field,  or  from  a  description  of  the  field  taken  from  the 
deed,  if  one  can  be  obtained.  From  the  deed  will  be  obtained 
only  the  description  of  the  boundary ;  and  the  other  features 
that  may  be  desired  must  be  sketched  in  the  field.  On  inspec- 
tion it  is  found  that  Mr.  Miller  owns  a  farm  bounded  on  one  side 
by  the  center  of  the  road,  on  another  by  the  Green  River,  and 
on  three  remaining  sides  by  fences,  broken  on  one  side  by  a 
pond,  which  is  owned  partly  by  Mr.  Miller  and  partly  by  his 
neighbor. 

But  one  convenient  way  to  get  the  area  of  the  field  is  known 
to  the  surveyor,  and  that  is  to  divide  the  field  into  triangles, 
measure  the  sides  of  the  triangles  or  two  sides  and  an  included 
angle  of  each,  whereupon  the  area  of  each  may  be  computed 
and  the  whole  summed.  It  is  evident  in  the  case  of  the  Mil- 
ler farm  that  what  might  be  considered  as  two  sides,  those 
formed  by  the  river,  are  not  straight,  and  therefore  can  not  be 
taken  as  sides  of  a  triangle.  Two  auxiliary  sides,  DE  and  EF, 
are  chosen,  lying  as  nearly  as  may  be  parallel  to  the  two  sides. 
It  is  found  impossible  in  the  field  to  choose  the  point  E  so  that 
A  may  be  seen  from  it,  hence  it  is  so  chosen  that  it  shall  lie  on 
a  perpendicular  to  the  line  A  F  drawn  through  F.  This  makes 
the  triangle  AFE  a  right-angled  triangle,  and  hence  AE  need 
not  be  measured.  If  the  house  were  so  located  that  the  line  BF 
could  be  conveniently  laid  out,  it  might  be  measured  and  the 
two  triangles  ABF  and  BFE  might  be  used  instead  of  AFE 
and  AEB. 


206 


LAND   SURVEYS. 


190.  Survey.  To  begin  the  survey,  select  a  point,  F,  in  the 
center  of  the  road,  and,  placing  a  flag  in  the  fence  at  B,  or 
merely  sighting  along  the  fence,  if  it  is  straight,  locate  a  point 
in  the  center  of  the  road  at  A,  in  Ba  prolonged.  Measure 
FA.  Measure  AB.  If  the  fence  aB  is  not  straight,  or  is  a, 


MAP  OF  FARM 

OF 

E.MILLER. 

PUYALLUP  CO.,  WASHINGTON, 

Kote:  Surveyed  by  A.B.Wooct  with  chain  nnltf 
Oct.i,  I8»t.  Scale;  2  chains  per  inch . 

FIG.  102. 


rough  rail  fence,  so  that  it  is  not  convenient  to  measure  on  the 
line  AB,  measure  along  a  parallel  line  offsetting  a  short  dis- 
tance from  the  fence,  making  no  note  of  the  offset,  but  record- 
ing merely  the  length  AB  and  the  distance  from  A  to  a.  Meas- 
ure along  the  line  BO.  or  on  a  small  offset  parallel  to  it  toward 
(7,  until  the  pond  is  reached.  Note  the  distance,  and  erect  a 


SURVEYING   WITH   THE   CHAIN.  207 

perpendicular  offset  long  enough  to  permit  a  line  parallel  to 
BO  to  pass  the  pond.  At  the  extremity  of  this  offset  erect  a 
perpendicular  which  will  be  parallel  to  BC,  and  measure  along 
the  perpendicular  far  enough  to  clear  the  pond,  and  then,  by  a 
process  like  that  just  used,  get  back  on  the  line  CB.  When 
measuring  along  the  parallel  line,  take  offsets  to  the  pond  as 
often  as  may  be  necessary,  noting  the  distance  to  each  offset 
arid  the  length  of  the  offset.  Having  reached  C,  measure 
along  the  line  CD  to  D,  then  along  DE  to  E,  noting  the  dis- 
tance to  and  the  length  of  each  offset  that  it  appears  necessary 
to  take  to  correctly  locate  the  river;  and  having  reached  E, 
measure  EF,  noting  similarly  the  offsets  and  also  the  distance 
to  the  fence  at  the  roadside. 

Perhaps  the  ordinary  method  of  procedure  now  would  be  to 
measure  the  diagonals  DB  and  BE ;  then  the  field  would  be 
divided  into  four  triangles,  ABE,  BCD,  BED,  and  AEF,  the 
areas  of  which  could  be  computed,  and  to  their  sum,  the  area 
between  the  lines  DE  and  EF  and  the  river  (computed  as 
described  in  Chapter  VI.)  could  be  added,  and  the  area  of  the 
entire  field  would  thus  be  obtained.  The  line  DB  would  be  a 
difficult  one  to  run,  because  both  D  and  B  being  lower  than  the 
ground  between  them,  the  line  must  be  ranged  out,  and,  more- 
over, the  intervening  woods  make  it  even  more  difficult  to 
determine.  If  the  woods  were  thick,  it  could  be  obtained  only 
by  running  a  trial  line  and  correcting  that  to  the  true  line. 
Moreover,  the  method  above  described  gives  no  check  on  the 
work ;  and  the  constant  thought  of  the  surveyor  should  be, 
"  Where  can  I  find  a  check  on  my  work  ?  "  Not  more  than  one 
check  on  one  piece  of  work  is,  however,  necessary  ;  though,  if 
more  may  be  obtained  without  waste  of  time,  they  may  some- 
times prove  advantageous. 

Having  measured  the  boundary  of  the  field,  it  is  probably 
best  now  to  measure  the  line  AC,  finding  by  trial  the  points 
at  which  perpendiculars  to  this  line  will  pass  through  F,  B,  E, 
and  D.  Measure  these  perpendiculars,  and  now  four  sides  of 
the  boundary  are  the  hypotenuses  of  right-angled  triangles, 
of  which  the  other  two  sides  are  known.  The  other  two  sides 
are  sides  of  trapezoids,  or  may  be  considered  the  hypotenuses 
of  right  triangles.  The  computed  value  of  each  of  the  sides  of 


208  LAND  SURVEYS. 

the  boundary  should  agree  with  its  measured  length.  If  it 
does  not,  there  is  an  error  in  the  measurement  of  either  the 
line  AC,  one  or  more  of  the  perpendiculars,  or  one  or  more  of 
the  sides  of  the  boundary.  If  all  is  right  except  one  side,  the 
error  is  in  that  side.  The  work  should  then  be  remeasured  in 
so  far  as  it  may  be  found  wrong.  If  this  latter  method  is  not 
adopted,  a  rough  check  may  be  had  on  the  former  method  by 
measuring  the  angles  made  by  the  sides.  These  should  meas- 
ure on  a  drawing  the  same  as  found  in  the  field. 

The  work  necessary  to  obtain  the  area  of  the  farm  is  now 
completed.  If  it  is  desired  to  locate  the  drives,  buildings,  and 
other  objects  within  the  inclosure,  it  may  be  done  by  running 
auxiliary  lines  at  known  angles  and  from  known  points  on  the 
various  sides.  For  instance,  a  line  could  be  run  from  E,  at 
right  angles  to  EF,  and  at  stated  points  on  this  auxiliary  line 
offsets  could  be  measured  to  the  corners  of  the  objects  it  is 
desired  to  locate ;  or  if  it  were  the  driveway,  the  offsets  could 
be  taken  at  frequent  noted  intervals  to  the  sides  of  the  drive. 
As  the  point  D  can  not  be  seen  from  the  line  A  0,  a  point  may 
be  chosen  at  random,  as  near  as  possible  to  the  proper  position, 
and  a  perpendicular  run  out  to  a  point  opposite  .Z),  whereupon 
the  length  of  the  perpendicular  and  the  distance  that  the  point 
on  AC  must  be  shifted  to  be  in  its  proper  place  become  known. 

191.  Notes.  The  measurements  should  be  written  on  the 
sketch,  which  must  be  made  large  enough  to  permit  this  to  be 
done  without  confusion.  It  is  believed  that  no  form  of  notes 
for  such  work  is  so  good  as  a  sketch  on  which  all  information 
is  written.  For  the  auxiliary  lines  locating  drives,  etc.,  the 
sketch  would  consist  of  a  straight  line,  with  distances  to  offsets 
marked  along  it,  and  offsets  and  objects  offsetted  to  sketched 
in  with  dimensions.  These  need  not  be  to  any  scale. 


FARM  SURVEYS. 

192.  Classes.  All  land  surveys  may  be  divided  into  three 
classes,  —  original  surveys,  resurveys,  and  location  surveys. 
These  surveys  are  made  with  a  compass  or  transit.  The  sur- 
veys of  former  years,  in  the  older  settled  portions  of  the  United 


FARM   SURVEYS.  209 

States,  were  all  made  with  the  compass,  and  almost  always  with 
reference  to  the  magnetic  meridian. 

193.  Original  surveys.     Original  surveys  are  those  made  for 
the  purpose  of  mapping  a"  field  whose  boundaries  are  marked  in 
some  way,  for  determining  its  area,  and  for  making  a  descrip- 
tion from  which  it  could  be  again  laid  out  if  the  boundaries 
should  be  destroyed.     Thus,  Mr.  Brown  may  own  a  farm,  a 
portion  of  which  is  timber  land,  which  Mr.   Black  wishes  to 
buy.     The  boundaries  are  sufficiently  marked  by  the  edges  of 
the  growth  of  timber.     After  the  timber  is   gone,  however, 
there  will  be  nothing  to  mark  the  boundaries,  unless  the  tract 
is  fenced.     There  is  no  way  in  which  Mr.  Black  may  know  how 
many  acres  he  is  to  buy,  unless  the  tract  is  surveyed.     Neither 
can  Mr.  Brown  give  Mr.  Black  a  deed  to  the  property  that 
would  contain  a  definite  description  from  which  the  tract  could 
be  laid  out  on  the  ground.     A  surveyor  is  called  in  and  shown 
the  tract  and  asked  to  make  a  survey  of  it,  to  compute  the 
acreage,  and  to  write  a  description  of  the  plot. 

194.  Making  an  original  survey.     At  the  corners  that  are 
shown  him  he  sets  monuments,  preferably  of  stone.     Too  many 
surveyors  set  merely  small  stakes,  which  soon  rot  or  are  pulled 
out.      These  monuments  are  "  witnessed "  by  trees,  or  other 
natural  objects  near  by,  whose  positions  relative  to  the  corner 
are  observed  and  noted.      The  witness  points  are  marked  as 
indicated   in  Art.  196.      The    surveyor  then   determines   the 
bearing   and   length   of   each  side,   usually  beginning  at  one 
corner  and  working  round  the  field  in  one  direction  till   he 
closes  on  the  corner  from  which  he  started.     He  keeps  notes  of 
the  work  in  a  notebook,  preferably  in  the  form  shown  on  page 
229,  or  on  a  sketch,  which  is  afterwards  "written  up"  in  the 
form  just  mentioned,  and  from  the  notes  he  "  tables  "  the  sur- 
vey and  computes  the  area  as  explained  in  Chapter  VI. 

195.  Making  the  map.     He  may,  and  usually  does,  make  a 
map  of  the  survey.     This  map  should  conform  to  the  require- 
ments stated  in  the  Appendix,  page  355.     The  easiest  and  most 
rapid  method  of  plotting  the  map  is  by  "  latitudes  and  longi- 
tudes."    Two  reference  lines  at  right  angles  are  drawn,  and 

K'M'B  simv.  — 14 


210  LAND  SURVEYS. 

one  is  assumed  as  the  meridian  and  the  other  as  a  line  of  zero 
latitude.  The  total  latitudes  and  longitudes  of  the  corners  are 
determined  from  the  tabling  work,  arid  are  laid  off  from  these 
base  lines.  The  latitudes  are  measured  from  the  zero  parallel, 
and  the  longitudes  from  the  meridian.  It  will  be  convenient 
to  make  the  reference  meridian  pass  through  the  most  westerly 
point  of  the  survey.  This  is  not  necessary.  To  avoid  negative 
signs  for  latitude,  compute  all  latitudes  as  if  the  reference  par- 
allel were  through  the  same  point  as  the  meridian ;  then  add 
to  all  latitudes  a  sum  equal  to  the  greatest  negative  latitude; 
or  assume  the  latitude  of  the  most  westerly  point  to  be  so  large 
that  there  shall  be  no  negative  latitudes. 

When  the  corners  are  all  plotted  they  are  connected  by 
right  lines,  and  the  outline  is  complete.  It  remains  to  number 
the  corners,  to  write  the  bearings  and  lengths  along  the  sides, 
and  to  put  on  the  necessary  descriptive  matter.  The  work 
should  all  be  done  with  the  utmost  neatness,  the  lettering  being 
preferably  the  simple  Roman,  the  most  effective  and  most  diffi- 
cult letter  that  is  made. 

196.  Description.  Having  made  the  map,  the  surveyor  writes 
a  description,  somewhat  in  the  following  fornr: 

Beginning  at  a  post  marked  B.I.,  at  the  S.E.  corner  of  the 
land  of  Joseph  Brown,  from  which  post  a  hard  maple  tree,  8 
inches  in  diameter,  bears  S.  10°  W.  10  links,  and  a  white  ash, 
12  inches  in  diameter,  bears  N.  70°  W.  50  links,  both  of  which 
trees  are  blazed  and  marked  B.l.B.T.  (Black  1  Bearing  Tree), 
and  running  thence  N.  10°  30'  W.  along  the  easterly  line  of 
said  Joseph  Brown,  six  and  forty-two  one-hundredths  (  6-j4^- ) 
chains  to  a  post  marked  B.2.,  from  which  post  a  hickory  tree, 
10  inches  in  diameter  and  marked  B.2.B.T.  bears  S.  68°  W. 
30  links  ;  thence  N.  84°  W.  seven  and  fifty  one-hundredths 
(7i5o°oO  chains  to  a  stone  about  12  inches  long  and  6  inches 
square,  set  flush  with  the  ground  and  marked  B.3.,  from  which 
stone  a  beech  tree,  15  inches  in  diameter  and  marked  B.3.B.T. 

bears,  etc., thence,  etc.,  to 

the  point  of  beginning,  containing acres,  more  or 

less. 

The  surveyor  usually  does  his  tabling  and  writes  his  descrip- 


FARM   SURVEYS.  211 

tions  in  a  book  which  he  keeps  for  the  purpose,  for  future 
reference.  Copies  are  made,  and,  with  a  tracing  of  the  map, 
are  furnished  the  person  for  whom  the  survey  is  made.  This 
is  the  simplest  kind  of  land  surveying. 

197.  Resurveys.     These  are  far  more  difficult  than  original 
surveys.      A  resurvey  consists  in  tracing  on  the  ground  an 
original  survey,  from  a  description  similar  to  that  just  given. 
The    difficulties  arise  from  the  destruction  of  monuments  and 
from   errors    in  original  work,  from  change  in  declination  of 
the  needle  (if  the  surveys  were  run  by  the  magnetic  compass 
and  referred  to  the  magnetic  meridian),  from  insufficient  data 
in  the  original  description,  such  as  failure  to  state  whether  the 
survey  is  referred  to  the  true  or  magnetic  meridian,  failure  to 
state  the  declination  on  which  the  survey  was  made,  lack  of 
bearing  trees  or  other  reference  points,  etc.,  and  from  conflict- 
ing testimony  of   interested   owners   as  to  where  the  corners 
were.     It  is  impossible  to  point  out  all   the    difficulties   that 
will  be  encountered  by  a  surveyor  in  his  attempt  to  reestab- 
lish the  monuments  of  an  old  survey.     The  more  of  this  work 
that  he  does,  the  more  firmly  will  it  be  impressed  upon  him 
that  the  only  kind  of  corners  to  establish  are  those  that  will  be 
as  nearly  as  possible  permanent,  and  that  minuteness  of  detail 
and  accuracy  in  all  descriptions  are  well  worth  the  time  they 
take. 

198.  Reasons  for  a  resurvey.    A  resurvey  becomes  necessary 
for  various  reasons.      Among  others  might  be  the  following, 
referring  to  the  original  survey  previously  described:    After 
Mr.  Black  has  owned  the  wood  lot  for  a  number  of  years,  and 
perhaps  has  cleared  it,  and  it  has  descended  to  his  son  and  his 
son's  son,  and  the  corners  are  mostly  obliterated,  it  is  sold  to 
Mr.  Johnson  who,  desiring  to  fence  it  off,  and,  moreover,  to 
see  whether  the  land  he  is  paying  for  is  all  there,  employs  a 
surveyor  to  "  run  it  out,"  giving  him  the  description  written  by 
the  original  surveyor  as  he  finds  it  in  his  deed,  with  possibly 
some  errors  in  copying. 

199.  Procedure.     If  the  surveyor  can  recover  a  single  line  of 
the  original  survey,  and  the  notes  are  correct,  his  work  will  be 


212  LAND  SURVEYS 

comparatively  easy.  (He  simply  has  to  establish  a  series  of 
lines  of  given  bearing  and  length.)  He  therefore  endeavors 
to  do  this.  If  this  can  not  be  done,  the  next  best  thing  is  to 
recover  any  two  corners  and  determine  in  the  field  the  bearing 
and  length  of  the  line  joining  them.  From  the  original  notes 
he  will  then  compute  the  bearing  and  length  of  this  line,  and 
the  angle  made  with  it  by  one  of  the  sides  of  the  field  joining 
it,  and  he  then  can  lay  off  this  angle  from  his  field-determined 
line  and  locate  that  side.  As  there  will  be  four  sides  joining 
the  line  between  the  two  corners,  it  will  be  seen  that  the  sur- 
veyor has  a  good  beginning  for  his  work. 

200.  Change  of  declination.  It  might  seem  that  if  he  could 
find  but  one  corner,  he  could  run  out  the  field,  knowing  the 
bearings.  This  would  be  true  if  he  knew  also  the  declination 
on  which  the  original  survey  was  made  and  the  declination  at 
the  time  of  his  resurvey. 

The  following  consideration  will  explain  this :  Let  it  be 
assumed  that  the  original  survey  was  made  with  reference  to 
the  magnetic  meridian  and  that  the  declination  at  that  time 
was  east  10°.  Then,  any  line  that  is  recorded  due  north  will 
be  10°  east  of  north.  Any  line  recorded  due  south  will  be  10° 
west  of  south,  etc.  That  is,  all  the  points  of  the  compass  are 
turned  10°  to  the  right.  Let  it  be  assumed  that  the  declina- 
'tion  at  the  time  of  the  resurvey  is  8°  east.  A  line  recorded 
as  due  north  will  be  8°  east  of  north,  and  a  line  recorded  as 
due  south  will  be  8°  west  of  south.  The  points  of  the  compass 
have  been  turned  back  two  degrees  to  the  left.  And  hence,  if 
a  line  were  run  out  with  the  original  bearing,  it  would  lie  two 
degrees  to  the  left  of  the  true  place.  It  would  be  necessary  to 
run  the  line  with  a  bearing  two  degrees  to  the  right  of  its 
originally  recorded  bearing.  Thus,  if  its  original  bearing  were 
north,  it  must  be  rerun  with  the  needle  reading  N.  2°  E. 
Hence,  the  following  rule  is  serviceable  : 

RULE  :  To  rerun  lines  recorded  by  their  magnetic  bearings, 
change  the  bearings  by  an  amount  equal  to  the  change  in  declina- 
tion and  in  a  direction  opposite  to  that  change  —  that  is,  left  or 
right. 

If  the  compass  has  a  declination  vernier,  that  vernier  may 


FARM   SURVEYS.  213 

be  set  so  that  the  compass,  when  the  line  of  sight  is  pointing  in 
a  known  bearing,  as  magnetic  north,  shall  read  a  bearing  as 
many  degrees  to  the  right  or  left  of  the  known  magnetic  bear- 
ing as  the  declination  has  changed  to  the  right  or  left.  It 
must  be  remembered  that,  in  using  the  declination  vernier,  the 
compass  box  moves  with  reference  to  the  line  of  sight.  In  the 
above  case  it  should  read  N.  2°  W.,  when  the  sights  are  point- 
ing to  magnetic  north.  Hence,  to  be  able  to  retrace  the  survey 
with  a  compass,  using  the  original  bearings,  the  following  rule 
is  serviceable  : 

RULE  :  Set  the  declination  vernier  so  that  the  compass  shall 
read  magnetic  bearings  erroneously  by  an  amount  equal  to  and  in 
the  direction  (left  or  right)  of  the  change  in  declination. 

To  find  the  change  in  declination,  if  it  is  not  known,  deter- 
mine the  bearing  of  a  line  of  the  original  survey  and  compare 
this  with  the  bearing  recorded.  The  difference  is  the  change 
in  declination  in  amount  and  in  opposite  direction  (left  or 
right). 

If  one  line  can  not  be  found,  but  two  corners  may  be,  con- 
nect the  two  corners  by  a  random  line  as  described  in  Art.  186. 
Find  the  bearing  of  the  closing  line  and  compare  it  with  the 
bearing  of  the  same  line  computed  from  the  original  notes. 

201.  Transit  or  compass.  When  a  transit  is  used,  or  in- 
deed a  compass,  the  angles  at  the  corners  of  the  field  may  be 
determined  from  the  recorded  bearings  and,  when  one  side  has 
been  recovered,  the  angles  and  distances  may  be  measured,  to 
recover  the  other  sides.  This  does  away  with  all  consideration 
of,  or  change  in,  declination.  It  is  by  far  the  better  method  to 
pursue  with  lands  of  any  considerable  value.  The  angles  de- 
termined from  compass  readings  can  not  be  depended  on  to 
minutes,  and  may  be  found  to  vary  five  or  ten  minutes.  It 
would  be  well  if,  when  a  transit  is  first  used  on  a  resurvey,  the 
corrected  bearings  and  distances  could  be  introduced  in  a  cor- 
rection deed  which  could  be  filed  with  the  proper  authority. 
If  angles  are  to  be  used,  and  one  side  can  not  be  directly  re- 
covered, but  a  closing  line  joining  two  corners  is  determined, 
proceed  as  follows:  Over  one  known  corner,  set  the  line  of 


214  LAND   SURVEYS. 

sight  in  the  closing  line  mentioned,  and  turn  off  an  angle  deter- 
mined, by  computation  from  the  description,  to  be  the  angle 
between  this  line  and  a  side  adjacent  to  the  corner  occupied. 

This  should  give  the  direction  of  that  side.  Measure  the 
recorded  length  of  that  side,  and  look  about  over  a  consider- 
able area  for  evidences  of  the  corner.  Look  for  recorded  wit- 
ness trees,  or  their  stumps.  Carefully  shovel  off  the  top  of  the 
soil  —  do  not  dig  it  up  by  spadefuls.  A  careful  shoveling  will 
frequently  reveal  the  hole  formerly  occupied  by  a  stake,  now 
rotted  entirely  away,  with  evidences  of  the  decayed  wood. 
Continue  the  work  till  all  corners  are  found  or  satisfactorily 
relocated.  The  only  thing  that  can  absolutely  insure  the  correct- 
ness of  the  resurvey  is  the  finding  of  the  old  corners. 

202.  Report.     Upon   the   completion  of   the  resurvey,  the 
surveyor  should  report  to  his  employer  just  what  he  finds.     It 
is  not  his  business  to  decide  controversies.     He  may  advise, 
just  as  an  attorney  would  do  ;  but  he  has  no  authority  to  cor- 
rect errors  or  to  establish  corrected  corners  as  the  corners.1 
His  business  is  to  make  an  examination,  to  reset  lost  corners  in 
their  original  positions,  if  he  can  find  them,  and  to  report  his 
method  of  procedure  and  the  reasons  for  his  action  to  his  em- 
ployer, who  may  then  take  such  action  as  he  chooses.     A  neat 
arid  explicit  map  should  accompany  the  report.     The  surveyor 
may  possibly  be  assisted  in  his  work  by  an  understanding  of  the 
principles  of  Art.  204,  deduced  from  many  court  decisions.2 

203.  Application  of   coordinates.     In   all   of  this  work   the 
coordinate  system  is  very  helpful.     Usually  the  true  corners 
of  a  tract  of  land  can  not  be  occupied  by  the  instrument,  nor 
can  the  lines  be  seen  throughout  their  length.     If  it  is  known 
where  the  corners  are,  and  it  is  desired  to  make  a  survey  for  a 
map  or  a  description,  a  random  survey  is  made  with  corners  as 
near  as  practicable  to  the  true  corners.  .  This  survey  is  bal- 
anced and  the  coordinates  of  its  corners  are  determined.     From 
such  of  the  corners  as  are  near  true  corners,  angles  or  azimuths, 

1  See  "Judicial  Functions  of  Surveyors,"  by  Judge  Cooley,  Appendix,  pages 
341-350. 

2  These  and  many  other  decisions  may  be  found  in  a  valuable    "  Manual  of 
Land  Surveying,"  by  Hodgman  and  Bellows. 


FARM   SURVEYS.  215 

and  distances  to  the  true  corners  are  noted,  and  from  these  the 
coordinates  of  the  true  corners  are  determined.  From  these 
coordinates,  lengths  and  bearings  of  true  lines  may  be  found. 

If  the  survey  is  for  the  purpose  of  relocating  lost  corners, 
and  the  bearings  of  the  lines  and  at  least  one  point  in  each  are 
known,  the  corners  may  be  located  even  though  the  lines  can 
not  be  occupied  by  the  instrument,  provided  the  corners  and 
known  points  are  accessible.  This  would  be  done  by  running 
a  random  survey  as  before  with  pointings  to  the  known  points 
(and  such  corners  as  are  known)  then  finding  the  coordinates 
of  the  known  points,  and  by  Problem  II.,  Art.  143,  the  coor- 
dinates of  the  wanting  corners.  Knowing  now  the  coordinates  of 
the  points  in  the  random  survey  and  those  of  the  true  corners, 
by  Problem  I.,  Art.  143,  find  the  bearing  and  distance  from  a 
corner  of  the  random  to  the  nearest  wanting  true  corner,  and 
locate  the  corner.1 

204.  Principles  for  guidance  in  resurveys.  I.  Construing  de- 
scriptions. The  following  principles  have  been  applied  to  the 
construing  of  descriptions  that  are  inconsistent,  obscure  in 
meaning,  or  imperfect. 

(1)  The    description  is   to  be  construed   favorably  to  the 
purchaser,  unless  the  intent  of  both  parties  can  be  certainly 
ascertained.     If  that  intent  can  be  ascertained,  the  description 
will  be  construed  accordingly. 

(2)  The  deed  must  be  construed  according  to  the  condi- 
tions existing,  and  in  the  light  of  the  facts  known  and  in  the 
minds  of  the  parties,  at  the  time  the  instrument  was  drawn. 

(3)  Every  requirement  of  a  description  must  be  met,  if  pos- 
sible.    Nothing  is  to  be  rejected  if  all  requirements  are  mutually 
consistent. 

(4)  If  some  parts  are  evidently  impossible  and,  by  rejecting 
such  parts,  the   remainder  forms  a  perfect  description,  such 
impossible  parts  may  be  rejected. 

(5)  A  deed  is  to  be  construed  so  as  to  make  it  effectual 
rather  than  void. 

1  For  model  example,  applied  to  city  property,  see  Appendix,  page  328.  The  stu- 
dent may  tell  how  to  proceed  if  the  bearings  and  lengths  are  all  given,  with  but  two 
corners  known,  and  lines  so  grown  over  or  occupied  by  structures  that  they  can  not 
be  run  out  directly  without  great  labor, 


216  LAND   SURVEYS. 

(6)  If  land  is  described  as  that  owned  and  occupied  by  an 
individual,  the  actual  line  of  occupation  is  a  requirement  or  call 
to  be  met  in  the  location. 

(7)  A   line  described  as  running  a  definite  distance  to  a 
definite  known  line  or  object,  will  be  construed  as  running  to 
that   object,    whatever   distance   is   required.     If    the   known 
object  is  uncertain  as  to  position,  the  written  distance  may  be 
used. 

(8)  The  terms  "  northerly,"  "  southerly,"  "  easterly,"  and 
"  westerly,"  are  to  be  construed,  in  the  absence  of  other  infor- 
mation, as  meaning  due  north,  south,  east,  and  west. 

(9)  When  a  definite  quantity  of  land  is  sold  and  nothing 
appears  to  indicate  its  form  —  as,  for  instance,  ten  acres  in  the 
northeast  corner  of  B's  land,  the  land  will  be  laid  out  as  a 
square,  unless  this  is  manifestly  impossible. 

(10)  A  description  by  "  metes  and  bounds  "  will  convey  all 
the  land  within  the  boundaries,  be  it  more  or  less  than  the  area 
mentioned  in  the  deed. 

(11)  Property  described  as  bounded  by  a  highway  extends 
to  the  center  of  the  highway,  unless  specifically  noted  other- 
wise. 

(12)  A   description    by  metes  and  bounds,  followed  by  a 
statement  that  the  land  described  is  a  particular  well-known 
parcel,  will   be  construed   to    convey  the  well-known   parcel, 
though    the   metes   and   bounds    do   not  fulfill  the  necessary 
conditions. 

II.  Water  boundaries.  (1)  Local  laws  of  different  states 
give  different  constructions  to  the  word  "  navigable "  and  the 
surveyor  must  examine  the  laws  of  the  state  in  which  he 
works.  The  United  States  statutes  provide  as  follows  for  the 
streams  within  the  area  known  as  the  public  lands  : 

"  All  navigable  rivers,  within  the  territory  occupied  by  the 
public  lands,  shall  remain  and  be  deemed  public  highways  ; 
and,  in  all  cases  where  the  opposite  banks  of  any  streams  not 
navigable  belong  to  different  persons,  the  stream  and  the  bed 
thereof  shall  become  common  to  both." 

(2)  Grants  of  land  bordering  on  navigable  streams  carry 
only  to  high-water  mark,  while  on  non-navigable  streams  they 
carry  to  the  center,  or  "filum  aquee." 


FARM   SURVEYS.  217 

(3)  The  common  law  holds  those  streams  only  to  be  navi- 
gable in  which  the  tide  ebbs  and  flows.      The  civil  law  con- 
siders a  stream  navigable  that  is  capable  of  being  used  as  a 
commercial  highway. 

The  courts  of  Pennsylvania,  North  Carolina,  South  Carolina, 
and  Alabama  follow  the  civil  law,  while  those  of  Maine,  New 
Hampshire,  Massachusetts,  Connecticut,  New  York,  Maryland, 
Virginia,  Ohio,  Illinois,  Indiana,  and  Michigan  follow  the 
common  law. 

(4)  The  bank  is  the  continuous  margin  where  vegetation 
ceases.     The   shore   is  the  sandy  space  between  it  and  low- 
water  mark. 

(5)  A  description  reading  "to  the  bank,"  or  "along  the 
bank,"  is  construed  to  mean  "the   bank,"  and  to  include  no 
portion  of  the  stream. 

(6)  Islands  in  rivers  fall  under  the  same  rule  as  the  land 
under  water,  and  belong  to  one  adjoining  proprietor  or  the 
other  —  unless    previously   lawfully   appropriated  —  according 
as  they  are  on  one  side  of  the  center  or  the  other.     The  filum 
aqute   is   midway  between  lines  of   ordinary  low-water  mark, 
without  regard  to  the  position  of  the  main  channel. 

(7)  Riparian   rights,  unless   expressly  limited,  extend   to 
the  middle  of  the  navigable  channel. 

(8)  In  some  states  the  tide  lands  are  held  to  belong  in- 
alienably to  the  people  of   the  state  and  may  not  be  sold  to 
individuals.     In  others  a  different  policy  has  been  pursued. 

(9)  A  boundary  by  the  shore  of  a  millpond  carries  to  low- 
water  mark. 

(10)  Boundary  lines  of  lots  fronting  on  a  river  extend  into 
the  river  at  right  angles  to  the  thread  of  the  stream,  without 
regard  to  the  form  of  the  bank. 

(11)  Land  made  by  the  drying  up  of  a  lake  or  the  deposit 
of  alluvium  along  a  river  accrues  to  the  adjacent  owners  and 
should  be  so  distributed  among  them  that  each  will  receive 
such  a  portion  of  the  made  area  as  his  former  frontage  on  the 
water  was  of  the  entire  former  frontage.     If,  however,   the 
water  front  is  the  valuable  item,  as  it  would  be  along  a  navi- 
gable river  or  lake  in  a  city,  the  new  frontage  is  to  be  dis- 
tributed according  to  the  old  frontages. 


218  LAND  SURVEYS. 

III.  Special  field  rules.  (1)  Monuments  control  courses 
and  distances.  That  is,  if  the  location  of  an  original  monu- 
ment can  be  certainly  ascertained  and  the  recorded  distance 
does  not  reach  that  monument,  the  line  must,  nevertheless,  be 
run  to  the  monument.  In  the  absence  of  sufficient  evidence  to 
determine  the  monument,  the  description  will  govern.  (The 
surveyor  should  use  every  effort  to  find  evidence  as  to  the 
location  of  the  lost  corner.) 

(2)  Adverse   possession  of   land  for  a  definite  period  of 
time  (varying  in  different  states),  even  without  color  of  title, 
constitutes  title  in  fee  ;  but  the  possession  must  be  adverse  ; 
that  is,  the  true  line  must  be  known  to  the  parties  or  the  line 
of  occupation  must  be  acquiesced   in   by  them.     If   the  true 
boundary  is  unknown,  and  each  claims  to  own  only  to  the  true 
line,  no  adverse  possession  can  arise. 

(3)  Boundaries  and   monuments   may  be   proved  by  any 
evidence  that  is  admissible  in  establishing  any  other  facts. 

(4)  A  resurvey  after  original  monuments  have  been  lost  is 
for  the  purpose  of  finding  where  they  were,  and  not  where  they 
should  have  been. 

(5)  Purchasers  of  town  lots  have  a  right  to  locate  them 
according  to  the  stakes  which  they  find  planted  and  recognized, 
and  no  subsequent  survey  can  be  allowed  to  unsettle  them. 
The  question  afterwards  is  not  where  they  should  have  been, 
but  where  they  were  planted  with  authority,  and   where  lots 
were  purchased  and  taken  possession  of  in  reliance  on  them. 

(6)  Of  two  surveys  that  disagree,  made  many  years  apart 
—  the  monuments  being  lost  —  the  original  survey  will  be  pre- 
ferred, particularly  if  the  line  of  the  first  survey  has  remained 
unquestioned  for  many  years. 

(7)  When  streets  have  been  opened  and  long  acquiesced 
in,  in  supposed  conformity  to  a  plot,  they  should  be  accepted 
as   fixed   monuments    in   locating    lots   or   blocks    contiguous 
thereto. 

(8)  A  beginning  corner  is  of  no  greater  dignity  or  impor- 
tance than  any  other  corner. 

(9)  A  call  for  a  lot  by  a  name  or  number  that  it  bears  on 
a  mentioned  plot  will  prevail  over  courses  and  distances,  and 
sometimes  over  monuments. 


UNITED   STATES   LAND  SURVEYS.  219 

205.  Location   surveys.     These  consist  in  laying  out  on  the 
ground  lines  previously  determined  by  computation  or  draw- 
ing.    Such  surveys  are  not  infrequently  connected  with  either 
original  surveys  or  resurveys.     Surveys  for  the  partition  and 
division  of  land  are  of  this  class.     If  Joseph  Brown  sells  to 
John  Black  five  acres  of  land  in  the  shape  of  a  square,  one 
corner  and  the  direction  of  one  side  being  fixed,  the  location  of 
the  sides  and  corners  would  be  a  location  survey.     Such  surveys 
are  comparatively  simple,  involving  sometimes  the  running  of 
one  or  more  trial  lines  for  data  to  compute  the  location.     Many 
complex  problems,  however,  arise. 

UNITED  STATES  PUBLIC  LAND  SURVEYS. 

206.  Value  and  character  of  work.      The  original  surveys 
for  the  subdivision  of  the  public  lands  of  the  United  States 
are  location  surveys.     The  method  adopted,  imperfectly  as  the 
work  has  been  done,  has  been  of  incalculable  value  in  definitely 
describing  each  separate  tract  or  parcel  of  land  sold  to  individ- 
uals, and  in  providing  that  lines  once  established  by  the  United 
States  deputy  surveyors  shall  remain  as  the  lines  they  purport 
to  be,  even  though  found  to  be  improperly  placed.     The  latter 
provision  has  been  of  particular  value  in  making  the  land  lines 
permanent.     The  work  of  subdividing  the  public  lands  is  almost 
completed,  and  hence  the  work  of  the  surveyor  of  the  future 
will  be  largely  resurveying,  that  is,  relocating  corners  that  have 
been  or  are  supposed  to  be  lost,  and  dividing  into  smaller  par- 
cels the  areas  already  located. 

If  the  original  surveys  had  been  properly  executed,  the 
work  would  not  be  difficult.  In  some  instances  the  work  was 
most  wretchedly  done,  either  willfully  or  through  ignorance ; 
and  the  corners,  when  established,  were  placed  far  from  their 
proper  positions.  The  notes  have  generally  been  returned  in 
a  form  indicating  correct  work  ;  and  hence  has  arisen  more  or 
less  difficulty  in  relocating  the  corners. 

207.  General  scheme  of  subdivision.     Certain   points  have 
been  selected  in  different  parts  of  the  country  through  which  true 
north  and  south  lines,  called  "  principal  meridians,"  have  been 
run  and  marked  out  on  the  ground.     Intersecting  these  prin- 


220  LAND   SURVEYS. 

cipal  meridians  at  the  initial  point  are  run  parallels  of  lati- 
tude, known  as  "base  lines."  On  either  side  of  these  meridians 
the  land  is  laid  out  in  approximately  square  parcels,  six  miles 
on  a  side,  called  "townships."  A  tier  of  these  townships  run- 
ning north  and  south  is  called  a  "range."  The  townships  are 

described  as  being  "  Township  No.  south  or  north  of  a 

named  base  line  and  range  No. east  or  west  of  a  named 

principal  meridian."  This  definitely  locates  every  township. 

The  lines  that  divide  the  ranges  are  called  "range  lines," 
and  those  that  divide  a  southern  from  a  northern  township  are 
called  "township  lines."  Each  township  is  further  divided 
into  thirty-six  "  sections,"  each  approximately  one  mile  square. 
These  sections  are  numbered  from  one  to  thirty-six.  Each 
section,  then,  is  definitely  located  by  its  number,  township,  and 
range.  The  sections  have  been  further  divided  into  quarter 
sections  as,  the  northeast  quarter,  the  southwest  quarter,  etc. 
Sometimes  the  sections  have  been  divided  into  halves,  described 
as  the  north  half  or  the  east  half,  etc.  The  Government  does 
not  divide  the  land  into  smaller  divisions  than  quarter  sections, 
but  it  sells  less  areas  than  this,  and,  in  such  cases,  and  when 
original  purchasers  sell  a  portion  of  their  purchase,  it  is  usual 
to  sell  a  quarter  of  a  quarter,  or  half  of  a  quarter,  or  even  a 
quarter  of  a  quarter  of  a  quarter  section  ;  and  the  method  of 
describing  these  fractional  portions  is  the  same  as  that  used  to 
describe  the  quarter  sections.  The  description  would  be  writ- 
ten as  follows  for  the  piece  described :  The  N.  E.  ^  of  the 
S.  W.  £  of  Sec.  26,  Tp.  8  N.,  R.  4  E.,  Mt.  Diablo  Meridian. 

The  positions  of  the  meridians  being  chosen  more  or  less  at 
random  and  at  different  times,  as  the  necessity  for  surveys  in 
different  localities  developed,  it  is  to  be  expected  that  the 
surveys  extending  east  from  one  meridian  will  not  close  with 
regular  full  sections  or  townships  on  the  surveys  extended 
west  from  the  next  easterly  meridian.  The  same  is  true  of 
tiers  of  townships  extended  north  and  south  from  adjacent  base 
lines.  The  result  is  fractional  townships  and  sections.  Some- 
times these  are  larger  than  the  standard  division,  sometimes 
smaller.  If  very  much  larger,  the  surplus  in  a  township  is 
divided  into  lots  which  are  numbered.  These  lots  are  made  to 
contain  as  nearly  as  possible  160  acres.  There  are  other 


UNITED   STATES   LAND   SURVEYS. 


221 


circumstances  that  will  appear,  that  cause  a  departure  from  the 
ordinary  method  of  subdividing  and  describing. 

The  work  in  a  given  state  is  under  the  direction  of  a 
United  States  surveyor  general.  The  work  has  all  been  done 
by  contract,  a  surveyor  taking  a  contract  to  perform  a  definite 
portion  of  work  for  a  specified  sum  per  mile.  This  has  been 
the  principal  cause  of  much  bad  work. 

The  lands  classified  as  public  lands  and  subdivided  accord- 
ing to  the  method  outlined  include  all  land  north  of  the  Ohio 
River  and  west  of  the  Mississippi  River,  except  Texas,  and 
including  Mississippi,  Alabama,  and  Florida,  except  in  all  the 
above-named  territory  such  lands  as  belonged  to  individuals  at 
the  time  the  territory  became  a  part  of  the  United  States. 

The  public  lands  of  Texas  are,  by  a  provision  of  the  laws 
admitting  Texas  into  the  Union,  the  property  of  the  state. 
No  general  scheme  for  w 

their  subdivision  -has  been 
developed.  They  have 
been  sold  in  parcels  as 
nearly  square  as  may  be. 
The  Spanish  vara  is  the 
unit  that  has  been  adopted  w 
for  measurement.  The 
vara  is,  in  Texas,  33^ 
inches.  There  are  5645 
square  varas  in  an  acre. 

The  following  is  very 

much  condensed  from  the  s 

"  Manual     of     Surveying  FlG-  103- 

Instructions  "  issued  by  the  General  Land  Office  in  Washington. 

208.  Historical  note.  The  first  surveying  of  the  public 
lands  was  done  in  Ohio  under  an  act  passed  by  Congress  in 
1785.  The  territory  included  in  this  early  survey  is  now 
known  as  "The  Seven  Ranges."  The  townships  were  divided 
into  thirty-six  sections  one  mile  square,  numbered  consecu- 
tively, as  in  Fig.  103. 

In  1796  the  method  of  numbering  sections  was  changed  to 
that  shown  in  Fig.  104 ,  and  this  method  is  still  in  use. 


36" 

30 

24" 

18 

12 

6 

35 

29 

23 

17 

11 

5 

34 

28 

22 

16 

10 

4 

33 

27 

21 

15 

9 

8 

32 

26 

20' 

14 

8 

2 

31 

25 

19 

13 

7 

1 

222 


LAND   SURVEYS. 


6 

5 

4 

3 

2 

1 

7 

8 

9 

10 

11 

12 

18 

17 

16 

15 

14 

13 

19 

20 

21 

22 

23 

24 

30 

29 

28 

27 

26 

25 

31 

32 

33 

34 

35 

36 

s 

FIG.  104. 

An  act  of  1805  directs  the  subdivision  of  public  lands  into 
quarter  sections,  and  provides  that  all  the  corners  marked  in 
the  public  surveys  shall,  be  established  as  the  proper  corners  of 
sections  which  they  were  intended  to  designate,  and  that  cor- 
ners of  half  and  quarter  sections  not  marked  shall  be  placed,  as 

nearly  as  possible,  "  equi- 
distant from  those  two 
corners  which  stand  on 
the  same  line."  This  act 
further  provides  that  "the 
boundary  lines  actually  run 
and  marked  .  .  .  shall  be 
established  as  the  proper 
boundary  lines  of  the  sec- 
tions or  subdivisions  for 
which  they  were  intended; 
and  the.  length  of  such 
lines  as  returned  by  .  .  . 
the  surveyors  shall  be  held 
and  considered  as  the  true  length  thereof.  ..." 

An  act  of  1824  provides  "  that  whenever,  in  the  opinion  of 
the  President  of  the  United  States,  a  departure  from  the  ordi- 
nary mode  of  surveying  land  on  any  river,  lake,  bayou,  or 
water  course  would  promote  the  public  interest,  he  may  direct 
the  surveyor  general  in  whose  district  such  land  is  situated 
...  to  cause  the  lands  thus  situated  to  be  surveyed  in  tracts 
of  two  acres  in  width,  fronting  on  any  river,  bayou,  lake,  or 
water  course,  and  running  back  the  depth  of  forty  acres."  l 

An  act  of  1820  provided  for  the  sale  of  public  lands  in  half- 
quarter  sections,  the  quarters  to  be  divided  by  lines  running 
north  and  south  ;  and  an  act  of  1832  provided  for  the  sale  of  the 
public  lands  in  quarter-quarter  sections,  and  that  the  half  sec- 
tions should  be  divided  by  lines  running  east  and  west.  The 
latter  act  also  provided  that  the  secretary  of  the  treasury  should 
establish  rules  for  the  subdivision  of  fractional  sections. 

209.  Legal  requirements  inconsistent.  Existing  law  requires 
that  the  public  lands  be  laid  out  in  townships  six  miles  square  by 

1  Let  the  student  determine  the  width  and  depth  in  chains  of  such  a  strip.  This 
provision  is  carried  out  where  the  water  front  rather  than  area  is  the  valuable  item. 


UNITED  STATES  LAND  SURVEYS.  223 

lines  running  due  north  and  south,  and  others  east  and  west, 
also  that  the  township  shall  be  divided  into  thirty-six  sections, 
by  two  sets  of  parallel  lines,  one  governed  by  true  meridians 
and  the  other  by  parallels  of  latitude,  the  latter  intersecting 
the  former  at  right  angles,  at  intervals  of  one  mile ;  and  each 
of  these  sections  must  contain,  as  nearly  as  possible,  six  hundred 
and  forty  acres.  These  requirements  are  manifestly  impossible 
because  of  the  convergency  of  the  meridians,  and  the  discrep- 
ancies will  be  the  greater  as  the  land  divided  is  farther  north. 
The  law  also  provides  that  the  work  of  subdivision  shall  be  so 
performed  as  to  throw  all  shortages  or  surplus  into  the  northern 
and  western  tiers  of  sections  in  each  township.  To  harmonize 
these  various  requirements  as  much  as  possible,  the  following 
methods  have  been  adopted  by  the  general  land  office. 

210.  Principal  reference  lines.     Initial  points  are  established 
astronomically  under   special   instructions.     From   the  initial 
point  a  "  principal  meridian"  is  run  north  and  south.    Through 
the  initial  point  a  "base  line"  is  run  as  a  parallel  of  latitude 
east  and  west.1     On  the  principal  meridian  and  base  line  the 
section  and  quarter  corners,  and  meander  corners  at  the  inter- 
section of  the  line  with  all  streams,  lakes,  or  bayous,  prescribed 
to  be  meandered,  will  be  established.    These  lines  may  be  run  by 
solar  instruments,  but  methods  involving  the  use  of  the  transit 
with  observations  on  Polaris  at  elongation  are  now  preferred.2 

211.  Standard  parallels.  —  Such   parallels,  called   also   cor- 
rection lines,  are  run  east  and  west  from  the  principal  merid- 
ian as  parallels  of  latitude  at  intervals  of   twenty-four  miles 
north  and  south  of  the  base  line.     "  Where  standard  parallels 
have  been  placed  at  intervals  of  thirty  or  thirty-six  miles,  re- 
gardless of  existing  instructions,  and  where  gross  irregularities 
require  additional  standard  lines,  from  which  to  initiate  new,  or 
upon  which  to  close  old,  surveys,  an  intermediate  correction  line 
should  be  established  to  which  a  local  name  may  be  given,  e.g., 
'Cedar  Creek  Correction  Line';  and  the  same  will  be  run,  in 
all  respects,  like  the  regular  standard  parallels." 

1  A  list  of  base  lines  and  principal  meridians  will  be  found  in  the  Appendix, 
pages  357-360. 

2  See  the  "  Manual  of  Surveying  Instructions"  for  detail  of  methods. 


224 


LAND  SURVEYS. 


212.  Guide  meridians.     Guide  meridians  are  extended  north 
from  the  base  line,  or  standard  parallels,  at  intervals  of  twenty- 
four  miles  east  and  west  from  the  principal  meridian. 

When  existing  conditions  require  the  guide  meridians  to 
be  run  south  from  a  standard  parallel  or  a  correction  line, 
they  are  initiated  at  properly  established  closing  corners 
on  the  given  parallel.  This  means  that  they  are  begun 
from  the  point  on  the  parallel  at  which  they  would  have 
met  it  if  they  had  been  run  north  from  the  next  southern 
parallel.  The  point  is  obtained  by  computation,  and  is  less 
than  twenty-four  miles  from  the  next  eastern  or  western 
meridian  by  the  convergence  of  the  meridians  in  twenty-four 
miles. 

In  case  guide  meridians  have  been  improperly  located  too 
far  apart,  auxiliary  meridians  may  be  run  from  standard  corners, 
and  these  may  be  designated  by  a  local  name,  e.g.,  "  Grass  Val- 
ley Guide  Meridian." 

213.  Angular  convergence  of  two  meridians.     This  is  given 

by  the  equation 

6  =  m  sin  L,  (1) 

where  m  is  the  angular  difference  in 
longitude  of  the  meridians,  and  L  is 
the  mean  latitude  of  the  north  and 
south  length  under  consideration. 
The  linear  convergence  in  a  given 
length  I  is 

c  =  I  sin  6.  (2) 

The  derivation  of  equation  (1)  is  as 
follows,  assuming  the  earth  to  be  a 
sphere,  which  will  introduce  no  error 
of  consequence  in  this  work.  In  the 
0  figure,  R  is  the  mean  radius,  r  is  the 
radius  of  a  parallel,  S  and  S'  are 
tangents  to  the  two  meridians.  The  angle  6  between  these 
is  the  angular  convergence  of  the  meridians  in  the  latitude  L. 
m  is  the  difference  in  longitude. 


UNITED  STATES  LAND   SURVEYS.  225 

r  =  R  cos  L.  (3) 

S  =  R  cot  L.  (4) 

S  and  S'  may  be  considered  as  radii  of  the  arc  AB,  which  has 
also  the  radius  r.     Since  a  given  length  of  arc  subtends  angles 
inversely  proportional  to  the  radii  with  which  it  may  be  drawn, 
6  =  r  _  R  cos  L  ,_. 

m~S      RcotL' 
whence  6  =  m  sin  L,  which  was  to  be  found. 

Since  the  distance  between  meridians  is  usually  given  in 
miles,  this  must  be  reduced  to  degrees.  This  is  done  by  first 
finding  the  linear  value  of  one  degree  for  the  mean  latitude, 
using  the  value  of  r  given  in  equation  (3).  To  make  (3) 
and  (4)  strictly  correct,  the  normal  at  A  should  be  used  instead 
of  R ;  but  the  mean  radius  will  give  results  sufficiently  close 
for  land  surveying.  See  Tables  IX.  and  X.,  pages  371,  372,  for 
values  of  6  and  length  of  1'  of  longitude  in  various  latitudes. 

214.  Township  exteriors.  Each  twenty-four  mile  "square" 
block  is,  when  practicable,  subdivided  into  townships  at  one 
time,  the  work  being  done  as  follows  : 

Beginning  with  the  southwestern  township,  the  meridional 
boundaries  or  range  lines  are  first  run,  and  on  these  are  estab- 
lished the  section  and  quarter  corners.  These  are  run  as  true 
meridians  from  south  to  north.  Next  the  east  and  west  lines, 
or  township  lines,  are  run  from  east  to  west  between  corre- 
sponding corners  of  the  range  lines.  On  each  such  township 
line  the  section  and  quarter  corners  are  established  at  full 
distances  from  the  eastern  range  line  of  each  range,  the  short- 
age being  thrown  into  the  most  westerly  half  mile  in  each 
range.  A  random  line  is  run  from  east  to  west,  temporary 
corners  being  set  at  correct  distances.  The  distance  north  or 
south  by  which  the  random  line  fails  to  reach  the  proper  corner 
is  observed,  and  from  this  and  the  known  length  and  bearing 
of  the  random  line  a  correct  bearing  is  determined  and  the 
correct  line  run  eastward,  on  which  are  placed  the  permanent 
corners.  If  a  random  line  fails  to  meet  the  required  corner  by 
more  than  three  chains,  it  must  be  rerun.  Deviations  from  the 
foregoing  methods  are  sometimes  made  necessary  by  the  topo- 
graphical features  of  the  territory. 

R'»rD  SURV. 15 


226 


LAND  SURVEYS. 


215.  Subdivision  of  townships.  Each  township  is  next  sub- 
divided into  sections  as  follows : 

Beginning  on  the  south  line  of  the  township,  at  the  corner 
common  to  sections  35  and  36,  run  northerly  parallel  to  the 
eastern  range  line  of  the  range  in  question,  one  mile,  setting  a 
quarter  corner  at  the  half  mile.  Establish  the  corner  common 

Toivnship  No.  5  North,  Range  No.  9  West,  of  a  Principal  Meridian 
Eat 


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T^«  above  plot  represents  a  theoretical  township  with  perfect  subdivisions, 
contiguous  to  the  north  side  of  a  Standard  Parallel  ;  in  assumed  Latitude 
hi  15'  N.,  and  Longitude  100°00'  W.  of  Gr.  Area  2S02U.16  A. 

FIG.  106. 


to  sections  25,  26,  35,  and  36,  and  run  east  on  a  random  line  to 
the  corner  common  to  sections  25  and  36  on  the  range  line,  set- 
ting at  the  half  mile  a  temporary  quarter  corner.  From  the 
observed  failure  to  meet  the  corner  on  the  range  line,  compute 
the  bearing  of  a  true  line,  and  run  this  from  the  corner  west, 


UNITED   STATES   LAND   SURVEYS.  227 

setting  the  permanent  quarter  corner  at  the  middle  point  of 
the  line. 

Proceed  thus  with  each  succeeding  section  to  the  north 
till  section  1  is  reached.  From  the  corner  common  to  sections 
1,  2,  11,  and  12,  the  meridional  line  is  run  to  the  section  corner 
on  the  next  township  line,  by  a  random  line  corrected  back  ;  but 
the  quarter  corner  is  set  at  forty  chains  from  the  south  end  so 
as  to  throw  whatever  discrepancy  there  may  be  into  the  north 
half  mile.  In  case  the  township  is  the  most  northerly  of  a 
twenty-four  mile  block,  the  line  between  sections  1  and  2  is  not 
run  to  the  corner  on  the  correction  line,  but  is  run  parallel  to  the 
range  line,  to  an  intersection  with  the  correction  line  where  a 
closing  corner  is  established  and  its  distance  from  the  section 
corner  noted. 

This  process  is  repeated  for  each  tier  of  sections  until  the 
fifth  tier  is  completed.  The  east  and  west  lines  of  the  sixth 
tier  are  run  from  east  to  west  as  random  lines  and  corrected 
back,  but  the  quarter  corner  is  set  just  forty  chains  from  the 
eastern  boundary  so  as  to  throw  the  discrepancies  into  the 
most  westerly  half  miles.  Fig.  106  represents  a  township  with 
perfect  subdivisions.  The  directions  written  along  the  various 
lines  indicate  the  directions  in  which  they  are  run.  Probably 
no  townships  that  have  been  surveyed  are  like  that  in  the 
figure,  all  of  them  being  more  or  less  distorted  by  inaccuracies, 
and  some  of  them  being  very  much  distorted.  When  very  bad 
work  is  discovered  in  time  to  correct  it,  before  such  alteration 
will  interfere  with  acquired  rights  of  individuals,  the  correc- 
tions may  be  made  under  rules  prescribed  by  the  General 
Land  Office. 

When  individuals  have  bought  the  land,  the  corners  as 
actually  set  must  remain  the  corners,  however  erroneously 
placed. 

216.  Meandering  a  stream.  This  consists  in  running  lines 
along  its  bank  to  determine  its  direction  and  length.  The  left 
and  right  banks  of  a  stream  refer  to  the  banks  as  they  would 
be  to  one  passing  down  stream. 

The  "  Instructions  "  provide  for  the  meandering  of  naviga- 
ble streams  and  those  whose  width  is  three  chains  and  upward. 


228  LAND   SURVEYS. 

They  are  to  be  meandered  on  both  banks.  Lakes,  deep  ponds, 
bayous,  etc.,  of  twenty-five  acres  or  more  are  also  meandered. 
Corners  called  "  meander  corners "  are  established  wherever  a 
meander  crosses  standard  base  lines,  township  lines,  or  section 
lines. 

A  meander  of  tidal  waters  follows  the  high  water  line. 

217.  Corners.     Stones,  posts,  trees,  and  earth  mounds  are 
used  to  mark  the  corners.     The  kind  of  corner  post  depends  on 
the  material  afforded  by  the  country.      The  "  Instructions " 
designate  the  kinds  of  corners  to  be  established  and  the  method 
of  marking  them. 

When  it  is  impossible  to  establish  a  corner  in  its  proper 
place,  auxiliary  corners  are  established  within  twenty  chains  of 
the  true  place,  on  each  of  the  lines  approaching  the  corner. 
These  corners  are  called  "  witness  corners."  The  center  quarter 
corner  of  a  section  is  located  by  connecting  the  opposite  north 
and  south  quarter  corners  by  a  straight  line,  and  placing  the 
center  corner  at  the  middle  of  this  line.  Exceptions  that  will 
suggest  themselves  are  necessary  in  the  northern  and  western 
tiers  of  sections. 

Any  person  having  to  do  with  surveys  of  lands  that  have 
been  subdivided  as  part  of  the  public  lands,  should  procure 
the  "  Instructions  "  and  study  them  carefully.  These  instruc- 
tions have  varied  from  time  to  time  and  it  is  well  to  consult 
those  in  force  at  the  time  the  original  surveys  in  hand  were 
made.1 

218.  Notes.     The  following  sample  page  of  the  field  notes 
prescribed  in  the  instructions  is  given  as  an  excellent  form  of 
notes  for  any  land  survey.     For  field  purposes  the  author  pre- 
fers a  sketch  on  which  all  notes  are  made,  arid  from  which  each 
night  the  field  book  may  be  written  up  in  the  form  given. 
Variations  in  items  noted  and  in  nomenclature  will  suggest 
themselves  to  the  maker  of  other  kinds  of  surveys. 

1  He  will  also  be  much  helped  by  a  book  entitled  "  Manual  of  Land 
Surveying"  by  Hodgman  and  Bellows.  This  work  contains  the  gist  of  many 
court  decisions  and  instructions,  resulting  from  many  years'  experience  in  this 
work.  Dorr's  "Surveyor's  Guide"  will  also  be  found  to  contain  many  practical 
suggestions. 


UNITED   STATES   LAND   SURVEYS. 


229 


GTH  GUIDE  MERIDIAN  EAST,  THROUGH  TPS.  13  N.,  BETWEEN  Rs. 
24  AND  25  E. 


Survey  commenced  August  29,  1890,  and  executed  with  a  W.  &  L.  E.  Gur- 
ley  light  mountain  transit,  No.  — ,  the  horizontal  limb  being  provided 
with  two  opposite  verniers  reading  to  SO"  of  arc. 

I  begin  at  the  Standard  Corner  of  Township  13  North,  Ranges  24  and  25 
East,  which  I  established  August  29,  1890.  Latitude  45°  34'.5  N.,  longi- 
tude 107°  24'  W. 

At  this  corner,  at  8h  54m  P.M.,  by  my  watch,  which  is  3"»  49»  fast  of  local 
mean  time,  I  observe  Polaris  at  eastern  elongation  in  accordance  with 
instructions  in  the  manual,  and  mark  the  point  in  the  line  thus  deter- 
mined by  a  tack  driven  in  a  wooden  plug  set  in  the  ground,  5.00  chs. 
north  of  my  station. 

August  29,  1890. 


August  30:  At  f>h  30™  A.M.,  I  lay  off  the  azimuth  of  Polaris,  l°49'.5to 
the  west,  and  mark  the  TRUE  MERIDIAN  thus  determined  by  a  cross  on 
a  stone  firmly  set  in  the  ground,  west  of  the  point  established  last  night. 
The  magnetic  bearing  of  the  true  meridian  is  N.  18°  05'  W.,  which  re- 
duced by  the  table  on  page  100  of  the  Manual  gives  the  mean  mag.  decl. 
18°  02'  E. 

From  the  standard  cor.  I  run 
North,  bet.  Sees.  31  and  36. 
Descend  over  ground  sloping  N.  W. 

Creek  10  Iks.  wide  in  ravine,  45  ft.  below  the  Tp.  cor.,  course  N.  32°  W. 
To  edge  of  table  land,  bears  N.  E.  and  S.  W. ;  thence  over  level  land. 
Bluff  bank,  bears  N.  58°  W.  and  S.  58°  E.  ;  descend  abruptly  40  ft. 
Bottom  of  ravine,  course  S.  58°  E. ;  ascend  50  feet  to 
Edge  of  table  land,  bears  S.  58"  E.  and  N.  58°  W. ;  thence  over  level  land. 
Difference  between  measurements  of  40.00  chs.,  by  two  sets  of  chainmen, 
is  18  Iks. ;  position  of  middle  point 
By  1st  set,  40.09  chs. 

By  2d  set,  39.91  chs.  ;  the  mean  of  which  is 

Set  a  limestone,  16  X  7  X  5  ins.,  11  ins.  in  the  ground,  for  J  sec.  cor., 
marked  4  on  W.  face,  and  raise  a  mound  of  stone,  2  ft.  base,  1J  ft. 
high,  W.  of  cor. 

Stream,  6  Iks.  wide,  in  ravine  15  ft.  deep,  course  N.  60°  W. 
Enter  heavy  oak  timber,  bears  E.  and  W. 

An  oak,  30  ins.  diam.,  on  line,  I  mark  with  2  notches  on  E.  and  W.  sides. 
Creek,  20  Iks.  wide,  1  ft.  deep,  course  N.  83°  W. 
Right  bank  of  creek,  begin  very  steep  rocky  ascent. 
Top  of  ridge,  250  ft.  above  creek,  bears  N.  80°  W.  and  S.  80°  E. 
Begin  descent. 

Difference  bet.  measurements  of  80.00  chs.,  by  two  chainmen,  is  22  Iks.  ; 
position  of  middle  point 
By  1st  set,  79.89  chs. 

By  2d  set,  80.11  chs.  ;  the  mean  of  which  is 
The  point  for  sec.  cor.,  150  ft.  below  top  of  ridge,  falls  on  a  flat  rock  in 

place,  10  ft.  E.  and  W.  by  6  ft.  N.  and  S.,  on  which  I 
Cut  a  cross  (X)  at  the  exact  cor.  point,  for  cor.  of  sees.  25,  30,  31,  and  36, 
marked  with  5  grooves  on  N.  and  1  groove  on  S.  sides  ;  from  which 
An  oak,  10  ins.  diam.,  bears  N.  22°  E.,  54  Iks.  dist.,  marked  T.  13 

N.,  R.  25  E.,  S.  30,  B.  T. 
A  dogwood,  5  ins.  diam.,  bears  S.  64J°  E.,  40  Iks.  dist.,  marked  T. 

13  N.,  R.  25E..S.  31,  B.  T. 
An  ash,  13  ins.  diam.,  bears  S.  51°  W.,  37  Iks.  dist.,  marked  T.  13 

N.,  R.  34  E.,  S.  36,  B.  T. 
An  oak,  9  ins.  in  diam.,  bears  N.  34"  \V.,  42  Iks.  dist.,  marked  T. 

13  N.,  R.  24  E.,  S.  25,  B.  T. 
Land,  level  and  mountainous. 
Soil,  gravel  and  rock  ;  4th  rate. 
Timber,  oak. 
Mountainous  or  heavily  timbered  land,  33.00  chs. 


230  LAND  SURVEYS. 


CITY  SURVEYING. 

219.  Precision  required.  Surveying  to  determine  land  lines 
in  a  town  or  city  does  not  differ  in  principle  from  surveying 
for  similar  purposes  in  the  country.  The  difference  lies  in  the 
degree  of  precision  required.  In  the  country  a  square  foot  of 
land  is  worth  from  one  cent  to,  perhaps,  in  extreme  cases,  five 
or  ten  cents.  In  the  heart  of  a  great  city  it  may  be  worth 
many  dollars,  and  a  single  inch  frontage  of  a  lot  one  hundred 
feet  deep  may  be  worth  several  hundred  dollars.  The  brick 
wall  of  a  modern  office  building  of  ten  or  mote  stories,  built  an 
inch  over  the  line  of  the  owner's  land,  would  be  a  costly  wall 
to  move,  and  an  error  so  locating  it  would  virtually  place  the 
owner  at  the  mercy  of  the  possessor  of  the  adjoining  property. 
It  is  therefore  readily  seen  that  an  error  in  closure  of  one  in 
three  hundred,  or  one  in  five  hundred,  which  may  be  tolerated 
in  farm  surveying,  would  be  altogether  out  of  the  question  in 
surveys  in  the  heart  of  a  great  city. 

In  such  situations  a  precision  of  one  in  fifty  thousand 
should  be  secured.  In  villages  and  small  towns  a  precision 
of  one  in  five  thousand  will  frequently  be  sufficiently  close, 
though,  if  the  place  has  "prospects,"  it  may  be  well  to 
make  closer  surveys.  The  methods  of  making  measure- 
ments to  the  various  degrees  of  precision  required  are  noted 
in  Chapter  I. 

To  lay  off  an  angle  so  that  the  position  of  a  line  may  not 
depart  .from  its  true  position  by  more  than  -5-5-5-5-^  of  its  length, 
requires  that  the  angle  have  no  greater  error  than  about  four 
seconds.  A  transit  reading  to  thirty  seconds  will  nearly  accom- 
plish this  if  the  angle  is  measured  three  times  before  reading, 
the  reading  being  divided  by  three. 

An  instrument  reading  to  twenty  seconds  will  ordinarily  do 
a  little  better  than  the  requirement,  and  an  instrument  read- 
ing to  ten  seconds  will  usually  accomplish  the  required  result 
with  a  single  measurement  of  the  angle,  though  it  should 
always  be  measured  more  than  once,  if  for  no  other  reason 
than  to  secure  a  check  on  the  work.  Of  course  the  magnify- 
ing power  of  the  telescope  must  correspond  to  the  fineness  of 
the  graduations. 


CITY   SURVEYING.  231 

It  has  been  stated  that  angles  are  read.  Bearings  are  not 
used,  nor  are  regular  traverses  run  out  in  city  surveys.  Bear- 
ings are  usually  worked  up  from  the  angles  for  all  the  lines  of 
a  survey,. in  order  to  compute  the  latitudes  and  longitudes  for 
determining  the  error  of  the  survey.  Azimuth  would  do  as  well. 

220.  Extent  of  survey.     City  land  surveys  are  usually,  ex- 
cept where  "  additions "  are  being  laid  out,  of  small  extent, 
covering  perhaps  a  single  lot  of  25  feet  by  100  feet,  more  or 
less.      In  a  well-monumented  city  the  entire  survey  may  be 
confined  to  the  block  in  which  the  lot  lies.     In  more  cases,  it 
will  be  necessary  to  go  several  blocks  away  to  obtain  monu- 
ments for  determining  the  necessary  lines.     In  cities  that  have 
been  laid  out  with  irregular  lines,  it  may  often  be  necessary  to 
carry  the  lines  and  measurements  over  the  roofs  of  the  solid 
blocks  of  buildings  in  order  to  determine  the  angle  points  in 
the  side  lines  of  the  lots.     Each  case  involves  new  difficulties 
that   the   surveyor   must   meet   by   exercising   his   ingenuity. 
Many  difficulties  are  solved  by  the  proper  use  of  the  coordinate 
system,  following  the  general  methods  outlined  in  Art.  203. 

221.  Instruments.     A  surveyor  who  undertakes  to  do  city 
work  should  be  supplied  with  the  best  instruments  for  meas- 
uring lines  and  angles.     The  tape  mentioned  in  Chapter  I.,  as 
used  in  New  York  City,  is  a  very  good  tool.     It  should  not  be 
used,  however,  until  it  has  been  tested  by  the  surveyor  himself 
to  verify  the  "  pull  scale  "  for  various  temperatures.     The  city 
surveyor  should  have,  in  his  office,  a  standard  length  marked 
out  on  the  floor,  or  some  other  place  not  subject  to  change,  by 
which  standard  he  may  test  his  tapes.    The  United  States  Coast 
and  Geodetic  Survey  Department  in  Washington,  D.C.,  will 
test  tapes  sent  there  for  that  purpose,  and  will  report  the  tem- 
perature, pull,  etc.,  for  which  they  are  standard,  and  their  con- 
stants.    For  this  service  a  very  small  fee  is  required. 

The  city  surveyor's  transit  should  be  of  high  magnifying 
power,  and  should  read  to  thirty  seconds  and  preferably  to 
twenty  seconds  or  even  ten  seconds.  It  should  be  well  made, 
and  of  a  pattern  to  insure  stiffness  and  permanency  of  adjust- 
ment. A  good  form  is  shown  in  Fig.  107. 


FIG.  107. 


CITY  SURVEYING.  233 

The  compass  being  of  little  use  in  city  surveying,  the  space 
usually  occupied  by  the  compass  box  is  used  to  form  the  base 
of  the  horseshoe-shaped  standards.  The  transit  is  made  with 
either  three  or  four  leveling  screws.  The  advantage  of  three 
screws  is  wider  leveling  base,  and  therefore,  with  given  pitch 
of  thread,  a  finer  adjustment  of  the  horizontal  plate  for  a  given 
turn  of  the  screw.  The  author  prefers  four  screws,  perhaps 
because  he  has  never  become  accustomed  to  the  use  of  three. 

222.  Description  of  a  city  lot.     The  description  of  a  city  lot 
as  found  in  a  deed  is  either  by  "lot  and  block"  or  by  "metes 
and  bounds." 

By  the  former  method  it  will  be  described  as  "Lot  No of 

Block  No of  the  original  survey  of  the  city  of " 

Or,  "Lot  No of  Block  No of. 's 

addition  to  the  city  of. as  the  same  is  shown 

and  delineated  on  a  certain  map  entitled    (Then  follows  the  title 

of  the  map)  filed  in  the  office  of  the  recorder  of 

county,  --^"Tiy--     (^y)__.      _._(y®sJl_." 

By  the  method  of  metes  and  bounds  it  will  be  described  as : 

"Beginning  at  a  point  in  the (N.,  S.,  E.,  or 

W.)  line  of. street  distant  thereon 

(Direction) (Distance)  feet  from  the  cor- 
ner formed  by  the  intersection  of  the  said line  of 

street,  and  running  thence  along  the  said 

line  of. street feet,  thence  at  an  an- 
gle  of degrees (Direction   written 

easterly,  northerly,  etc.) feet ;  thence  at  an  angle 

of degrees (Direction) feet ; 

thence  at  an  angle  of. degrees (Direc- 
tion)  feet  to  the  point  of  beginning,"     To  this  will 

sometimes  be  added:  "being  Lot  No of  block  No ," 

etc.,  as  was  written  above. 

223.  Finding  a  city  lot.     If  the  description  is  by  lot  and 
block,  it  will  be  necessary  to  refer  to  the  map  of  the  survey 
mentioned  for  the  data  as  to  the  widths  of  streets,  angles  in  the 
lines,  and  positions  of  lots,  before  the  survey  can  be  begun. 
The  map  should  also  show  the  character  and  position  of  mon- 
uments located  at  the  time  of  the  original  subdivision  of  the 


234  LAND   SURVEYS. 

tract.     This  it  will  rarely  do  if  the  survey  is  an.  old  one,  and 
it  may  not  do  so  even  if  the  survey  is  a  late  one.1 

It  will  often  happen  that  even  though  the  map  shows  the 
positions  of  the  monuments  that  originally  marked  the  lines, 
those  monuments  are  not  now  to  be  found.  In  such  cases,  the 
best  that  can  be  done  is  to  find  any  monuments  having  any 
bearing  on  the  survey  in  question,  and  further  to  note  existing 
lines  of  permanent  improvements  that  have  been  long  in  place, 
and  to  endeavor  to  reconcile  these  with  the  figures  shown  on 
the  map.  This  work  requires  the  best  judgment  of  the  sur- 
veyor, for  it  will  almost  always  be  found  that  the  discrepancies 
are  considerable.  This  is  not  so  true  in  the  centers  of  very 
large  cities,  for  here  surveys  have  been  gone  over  and  over  by 
most  careful  workmen,  and  the  lines  have  become  pretty  defi- 
nitely fixed  ;  but  in  smaller  and  newer  cities  there  is  an  endless 
amount  of  difficulty. 

224.  Marking  corners.     When   the   boundaries  of   the  re- 
quired lot  are  finally  fixed  upon,  each  corner  is  marked.     If  the 
ground  is  open,  a  stake  with  a  small  tack  may  mark  each  corner, 
if  the  marking  is  for  temporary  use. 

The  marking  may  be  a  mark  made  on  a  building  or  in  the 
stone  flagging  of  the  sidewalk,  or  otherwise.  A  sketch  of  the 
lot  with  a  note  of  the  marks  made  and  a  certificate  of  survey 
should  be  furnished  the  person  for  whom  the  survey  is  under- 
taken. 

225.  Discrepancies.     If   any   discrepancies   are    discovered, 
they  should   not   be  "fudged   in"  and   hidden,   but   mention 
should  be  made  of  just  what  is  found.      If  discrepancies  are 
found,  the  surveyor's  work  is  doubled,  for  he  must  be  sure  the 
apparent  errors  are  not  in  his  own  work.     If  the  discrepancies 
found  are  small,  no  greater  than  the  surveyor  would  expect  to 
find  in  repeating  his  own  work  —  that  is,  if  they  are  within  the 
limit  of  precision  required — no  note  need  be  made  of  them. 
When,  for   instance,  a  block   that   is   recorded  500  feet  long 
is  found  to  be  500.04  feet  long,  and  it  is  required  to  lay  out 
a  lot  in  any  portion  of  the  block,  the  measurement  from  one 
end  of  the  block  to  locate  the  lot  should  be  to  the  recorded 

i  Consult  Appendix,  page  354. 


CITY  SURVEYING.  235 

measurement  as  500.04  is  to  500.  Such  discrepancies  may  be 
thus  distributed. 

If  the  street  lines  on  both  sides  of  the  block  are  well  de- 
nned, and  the  block  is  found  to  overrun  or  fall  short  of  the 
recorded  length,  and  if  the  description  of  the  lots  sold  has 
been  only  by  lot  and  block,  the  error  should  ordinarily  be  di- 
vided proportionally  among  the  various  lots.  If  the  descrip- 
tion is  by  metes  and  bounds,  it  is  more  difficult  to  say  what  to 
do,  and  the  best  the  surveyor  can  do  is  to  report  what  he  finds 
and  ask  his  employer  for  further  instructions.  He  must,  in  no 
case,  locate  the  lot  in  what  he  thinks  should  be  its  proper  place 
and  attempt  to  defend  such  an  action  as  the  only  proper  course. 
Such  procedure  has  proved  a  source  of  much  litigation  for  the 
owner  and  loss  of  reputation  to  the  surveyor. 

If  the  description  is  by  metes  and  bounds  only,  the  surveyor 
must  determine  the  starting  point  by  finding  the  street  line  on 
which  it  lies  and  the  other  street  line  which,  with  the  former, 
forms  the  intersection  from  which  the  beginning  point  is 
located.  This  is  frequently  difficult  to  do  because  of  lack  of 
monuments  or  marks.  When  it  is  accomplished,  the  surveyor's 
work  is  clear  :  he  follows  the  description.  If  he  finds  discrep- 
ancies, he  reports  what  he  finds.  If  the  description  is  both  by 
metes  and  bounds  and  by  block  and  lot,  it  is  more  difficult 
properly  to  locate  discrepancies,  and  such  as  are  found  must  be 
reported.  The  interpretation  of  the  deed  would  perhaps 
usually  be  the  intent  of  the  buyer  and  seller,  if  that  could 
be  discovered.  The  surveyor  must  remember  that  he  has  no 
right  to  interpret  the  deed.  That  is  the  business  of  a  court. 
Descriptions  in  deeds  are  often  inconsistent  in  themselves  and 
indicate  impossible  plots. 

226.  Planning  additions.  Whenever  possible,  additions 
should  be  so  laid  out  that  the  streets  may  be  continuations  of 
those  in  the  adjacent  subdivided  portion  of  the  city.  Not 
enough  consideration  has  been  given  in  the  past  to  this  impor- 
tant feature.  Unless  the  ground  is  very  irregular  and  the 
addition  is  a  remote  suburb,  the  subdivision  should  be  rectan- 
gular. If  the  ground  is  irregular  and  on  the  extreme  outskirts 
of  the  city,  in  a  part  that  will  probably  never  be  thickly  built 


236  LAND   SURVEYS. 

up,  it  may  be  laid  out  in  curved  lines  to  conform  more  nearly 
to  the  surface  of  the  ground,  involving  less  work  in  grading 
streets  and  lots,  and  making  the  tract  more  beautiful.  In  such 
cases  the  tract  will  not  be  cut  up  into  very  small  lots.  No  lot 
should  have  less  than  one  hundred  feet  front,  and  it  maybe 
said  that  such  subdivision  should  not  ordinarily  be  undertaken 
if  the  lots  are  to  be  less  than  one  acre  in  area. 

In  planning  a  curved  subdivision,  the  streets  should  usu- 
ally be  located  so  as  to  give  good  drainage  to  the  lots.  This 
will  generally  mean  that  the  streets  will  occupy  the  lower 
ground.  To  do  this  work  most  satisfactorily,  a  contour  map 
of  the  tract  is  desirable.  This  is  made  as  described  in  Chap- 
ter IX. 

227.  Making  the  survey  and  map  for  an  addition.  The  first 
step  in  subdividing  an  addition  is  to  make  a  complete  survey  of 
the  entire  tract,  using  the  same  care  as  for  city  work.  Many 
discrepancies  of  greater  or  less  amounts  will  be  found.  The 
descriptions  of  the  tract  are  probably  from  old  farm  surveys. 
If  no  greater  discrepancies  are  found  than  might  be  expected 
from  such  work,  and  the  corners  are  well  established,  no 
trouble  need  arise.  The  new  distances  and  angles  are  the 
ones  that  will  be  recorded  on  the  map  that  is  to  be  made. 
The  surveyor  will  carefully  note  the  intersection  of  the  bound- 
ary with  the  street  lines  of  the  adjoining  tract,  and  the  direc- 
tions of  those  lines.  The  survey  will  be  very  carefully 
balanced  and  mapped. 

Probably  the  best  method  of  making  the  original  map  is 
to  divide  the  sheet  into  squares  of,  say,  one  hundred  to  one 
thousand  units  on  a  side  —  the  number  will  depend  on  the  size 
and  scale  of  the  map.  One  of  the  lines  of  division  will  be 
assumed  as  the  reference  meridian,  and  another,  at  right  angles, 
as  the  base  parallel.  Their  intersection  may  be  taken  as  the 
origin  or  may  be  given  any  arbitrary  number  of  hundreds  as  its 
coordinates.  Each  of  the  corners  of  the  squares  will  then  have 
for  its  ordinates  quantities  expressed  in  whole  hundreds.  The 
coordinates  of  any  point  in  the  survey  being  determined,  it  is 
at  once  known  what  square  the  point  falls  in,  and  the  point 
may  be  measured  in  from  a  corner  of  that  square,  and  be  cer- 


CITY  SURVEYING.  237 

tainly  within  reach  of  a  single  rule  length.  If  the  map  is  small, 
this  method  has  no  advantage  over  a  single  meridian  on  which 
latitude  ordinates  are  measured  and  perpendicular  to  which 
longitude  ordinates  are  measured. 

A  system  of  subdivision  will  then  be  planned  and  drawn 
on  the  map,1  and  then  located  on  the  ground.  The  data  for 
placing  the  street  lines  on  the  ground  will  not  be  scaled  from 
the  map,  but  will  always  be  computed.  The  coordinate  system 
is  recommended  for  this  work. 

The  location  on  the  ground  consists  in  placing  monuments 
at  all  intersections  of  street  lines.  These  are  sometimes  placed 
at  the  intersection  of  center  lines  and  sometimes  at  the  block 
corners.  Neither  practice  is  good.  Probably  the  best  place 
to  locate  them  is  in  the  sidewalk  area,  at  a  given  offset,  say  five 
feet,  from  the  property  line.  They  are  less  likely  to  be  dis- 
turbed here  than  elsewhere.  If  the  addition  is  to  be  laid  out 
on  curved  lines,  the  monuments  should  mark  the  intersections 
of  street  lines  and  the  points  on  the  arcs  where  the  radii  change. 
For  methods  of  locating  curves,  see  Chapter  VIII. 

The  monuments  should  be  of  stone,  durable  in  quality  and 
about  two  feet  long,  with  the  top  dressed  to  about  six  inches 
square  and  the  bottom  left  rough.  The  top  face  should  be 
smooth,  with  a  hole  drilled  in  its  center,  in  which  is  set  with 
lead  a  copper  bolt  with  a  cross  marked  in  its  head.  The  stone 
should  be  set  below  the  disturbing  action  of  frost.  A  piece  of 
iron  pipe  may  reach  to  the  surface  of  the  ground  and  terminate 
with  a  cast  cover.  Where  stone  is  costly,  concrete  may  be 
used,  formed  of  sand  and  cement,  inside  of  a  piece  of  stove  pipe. 
The  copper  bolt  should  be  set  as  before.  Other  forms  may 
suggest  themselves  to  the  surveyor. 

A  carefully  constructed  map  finishes  the  work.  This  map 
should  show  the  position  of  each  monument,  preferably  by  giv- 
ing its  coordinates  referred  to  some  origin,  the  relation  of  the 
monuments  to  the  property  lines,  and  such  further  information 
as  is  indicated  in  the  Appendix,  page  355.  If  curves  are  used, 
the  coordinates  of  the  centers  and  beginning  and  ending  points 
of  the  curves  should  be  noted. 

1  See  an  excellent  paper  by  J.  Stiibben,  of  Cologne,  Germany,  in  Vol.  XXIX., 
"  Transactions  American  Society  Civil  Engineers." 


CHAPTER  VIII. 


CURVES. 

228.  Use  of  curves.     When  a   road,  railroad,  or   canal   is 
built,  the  center  line  is  laid  out  on  the  ground.     The  center 
line  is,  in  the  case  of   a  railroad,  always  a  series   of   straight 
lines  connected  by  arcs  of  circles  to  which  the  straight  lines 
are  tangent.     In  the  case  of  a  canal  or  wagon  road,  this  is  not 

always  true,  as  the 
line  is  sometimes 
irregular,  following 
more  closely  the  con- 
formation of  the 
ground  than  does  a 
railroad.  It  would 
be  better  if  canals 
and  wagon  roads 
were  always  laid  out 
as  railroads  are.  In 
the  case  of  wagon 
roads,  the  radii  of 
connecting  curves 
may,  of  course,  be 
short.  Many  park 
roads  and  suburban  streets  are  located  as  arcs  of  circles  and 
straight  line  tangents.  Park  drive  curves  are  often  irregular 
and  sometimes  are  successive  arcs  of  varying  radii,  known  as 
compound  curves.  Railroad  curves  are  frequently  compound 
curves. 

229.  Principles.     Only  the  fundamental   principles  of  the 
laying  out  of  these  curves  will  be  here  given.     The  straight 


PRINCIPLES   OF   CURVES.  239 

lines  are  called  technically  "tangents."  The  curves  are  known 
by  the  angle  subtended  at  the  center  by  a  chord  of  one  hundred 
feet.  Thus,  if  such  a  chord  subtends  4°,  the  curve  is  known 
as  a  4°  curve.  Any  two  tangents  not  parallel  intersect  at 
some  point,  and  the  angle  at  the  point,  between  one  tangent 
produced  and  the  other  tangent,  measured  in  the  direction  in 
which  the  line  is  to  bend,  is  known  as  the  intersection  angle. 
Curves  are  measured  in  chords  of  one  hundred  feet.  This  dis- 
tance is  a  "  station,"  whether  used  on  curve  or  tangent. 

A  curve  four  hundred  feet  long  means  a  curve  of  four 
stations.  It  does  not  mean  that  the  arc  is  just  four  hun- 
dred feet  long,  but  that  there  are  four  chords,  each  of  one 
hundred  feet.  In  Fig.  108,  two  tangents,  A  V  and  VB,  are 
connected  with  a  curve  of  radius  R.  AB  is  known  as  the  long 
chord,  O.  DE  is  the  middle  ordinate,  M.  AV=VB  is  the 
tangent  distance,  T.  VE  is  the  external  distance,  E.  From 
the  figure  it  is  evident  that 

(1)  Jf=72versiA,       (3) 

(2)  ^=J2exsecJA.    (4) 

External  secant  is  a  trigonometrical  function  not  commonly 
mentioned  in  Trigonometry,  but  very  useful  in  handling  curves. 
It  is  the  "secant  minus  one."  In  the  above  equations  it  will 
be  seen  that  the  radius  R  is  used,  while  it  has  been  said  that 
curves  are  known  by  their  "degrees."  If  AB  is  one  hundred 
feet,  A  becomes  D,  the  degree  of  the  curve,  and  from  (2) 

7?         50  _ 

jB  =  sh^p'  <5) 

For  any  other  curve  of  degree  D1  ', 


For  small  angles  it  may  be  assumed  that  the  sines  vary  as  the 
angles  ;  whence 

R        D'  sin  1°      D'  ~ 

~      ' 


R' 

The  radius  of  a  1°  curve  is  5729.65  feet.     Except  in  close  land 


240  CURVES. 

surveys  or  surveys  of  city  streets,  the  radius  may  be  taken  as 
5730.00  feet.  In  using  curves  in  land  surveys,  the  term  "de- 
gree" should  be  abandoned  and  all  circles  designated  by  their 
radii.  Since  the  values  of  all  the  functions  given  in  equations 
(1),  (2),  (3),  (4),  vary  directly  as  R,  A  remaining  constant, 
these  functions  will  all,  assuming  the  correctness  of  equation 
(7),  vary  inversely  as  D.  They  would  vary  exactly  in  this 
way  if  the  one  hundred  feet  that  measures  D  were  measured 
on  the  arc  instead  of  as  a  chord.  This  will  be  clear  from  the 
principles  of  Geometry.  The  length  of  a  curve  for  a  given 
value  of  A  and  D  is  in  stations, 

'         ff  =  |  (8) 

and  in  feet  measured  in  chords  of  one  hundred  feet, 

£=100--  (9) 

It  is  frequently  required  in  practice,  to  know  the  values  T 
and  E  for  given  or  assumed  values  of  A  and  D.  If  a  table  is 
made  of  values  of  these  quantities  for  a  great  number  of  val- 
ues of  A  and  for  D  =  1°,  the  value  of  T  or  E  for  any  other 
value  of  D  would  be  at  once  obtained  by  dividing  the  value 
found  in  the  table  opposite  the  given  A,  by  the  given  value 
of  T>.  In  practical  railroad  work  this  method  may  be  used 
for  curves  of  less  than  8°  or  10°  without  serious  error.  For 
sharper  curves  than  these  the  quantities  should  be  computed 
by  equations  (1),  (2),  (3),  and  (4).  If,  however,  the  sharper 
curves  are  measured  in  fifty-foot  chords,  up  to  say  14°  to  16° 
curves,  and  in  twenty-five-foot  chords  thereafter,  the  approxi- 
mate method  may  be  used  up  to  at  least  20°  curves  or  24° 
curves  without  serious  error.  This  means  that  a  14°  curve 
will  be  defined  as  one  in  which  a  chord  of  fifty  feet  subtends 
at  the  center  an  angle  of  7°.  A  20°  curve  is  one  in  which  a 
twenty-five-foot  chord  subtends  an  angle  of  5°  ;  etc. 

230.  Laying  out  the  curve.  The  point  at  which  the  curve 
is  to  begin  is  known  when  the  point  V  and  the  tangent  distance 
T  are  known.  For  the  latter,  A  and  D  must  be  known.  Let 
the  point  of  beginning,  called  the  P.  C.  (point  of  curve),  be  A. 
A  transit  is  set  at  A,  Fig.  109,  an  angle  =  |  D  turned  from  the 


LAYING   OUT   CURVES. 


241 


tangent  A  V  to  C.     The  chain  or  tape  is  stretched  from  A  and 
the  farther  end  made  to  coincide  with  the  line  of  sight  at  C.1 

From  the  line  CA  an  angle  equal  to  |  D  is  laid  off,  and  the 
chain  stretched 
from  (7  and  the  far- 
ther  end  put  in  the 
line  of  sight  at  D. 
Thus  the  points 
one  hundred  feet 
apart  are  located. 
The  final  deflection 
angle  VAB  is  g  A. 
If  the  curve  is  not 
of  a  length  ex- 
pressed by  a  whole 
number  of  stations, 
there  must  be 
turned  off  for  the 
"subchord  "  a  de- 
flection usually  made  proportional  to  the  length  of  the  sub- 
chord  ;  thus  in  the  figure,  if  EB  is  fifty  feet,  the  deflection 
EAB  is  one  half  of  l  D,  etc.  This  again  assumes  that  the 
curve  is  measured  on  the  arc,  and  the  method  is  practically 
correct  for  curves  of  less  than  10°.  For  sharper  curves  than 
this  a  computation  should  be  made.  Thus,  since  A  is  known, 
there  remains  to  turn  off  from  AE  an  angle  equal  to  \  A  —  VAE. 
This  being  half  the  angle  subtended  by  the  chord  EB,  that  chord 
may  be  computed.  If  the  curves  of  higher  degree  are  laid  out 
with  short  chords  as  before  described,  the  proportionality  of 
chord  to  deflection  angle  may  still  be  assumed. 

If  the  whole  of  the  curve  can  not  be  seen  from  the  beginning, 
the  transit  may  be  moved  to  a  point  on  the  curve  already 
located  and  the  work  continued  as  follows:  Let  it  be  assumed 
in  Fig.  109  that  there  is  an  obstruction  in  the  line  AE  so  that 
E  can  not  be  seen  from  A.  Move  the  transit  to  D,  and  with 
the  verniers  set  at  zero,  with  the  lower  motion,  turn  the  line  of 
collimation  on  A.  Now  lay  off  to  the  right  an  angle  equal  to 
D  (the  deflection  from  A  to  7))  and  the  line  of  collimation  will 

1  Let  the  student  show  that  this  procedure  locates  the  point  C  on  the  curve. 
R'M'D  SURV.  — 16 


242 


CURVES. 


lie  in  the  tangent  to  the  curve  at  D.  Transit  the  telescope 
and  deflect  to  the  right  for  the  station  E,  etc.,  as  usual.  Not 
much  more  than  90°  of  an  arc  should  be  located  from  one 
point.1 

Points  occupied  by  the  transit  are  usually  solid  "  hubs  " 
driven  flush  with  the  ground  with  a  small  tack  to  mark  the 
exact  point.  The  other  points  are,  in  railroad  and  common 
road  surveys,  merely  marked  with  stakes.  In  close  land  sur- 
veys and  city  work,  each  point  should  be  marked  with  hub  and 
tack. 

231.   Location  by  chain  alone.     A  method  for  locating  arcs 
approximately  by  the  use  of  the  chain  or  tape  is  as  follows, 
and  is  known  as  the  method  by  "  chord  off- 
sets "  : 

In  Fig.  110,  A  V  is  a  tangent  to  which  a 
curve  of  degree  D  is  to  be  joined  at  A.  If 
AS  is  one  station, 

£5  =  100  sin  ID. 

Imagine  the  curve  continued  back  one  sta- 
tion to  E.  eE  =  £b,  and  /,  in  EA  produced, 
is  distant  from  b  by  an  equal  amount,  therefore 


Bb  is  known  as  the  tangent  offset  t,  and  fB  is 
the  chord  offset  =  2  t.  To  locate  the  curve, 
swing  the  chain  from  A  and  place  the  farther 
end  of  it  a  distance  t  from  the  tangent.  Stretch 
the  chain  from  B  in  line  with  AB  and  make  a 
mark  at  <?.  Swing  the  chain  about  B  till  cO 
equals  2  1.  Proceed  in  the  same  manner  for 
the  remainder  of  the  curve.  If  the  curve  begins  with  a  frac- 
tional station,  the  tangent  offset  for  the  fractional  station  may 
be  computed  and  laid  off  ;  and  for  the  remainder  of  a  full  sta- 
tion back  of  A,  the  same  thing  may  be  done.  The  direction  of 
the  chord  of  a  full  station  on  the  curve  is  now  determined,  and 
the  work  may  proceed  as  usual. 


1  Let  the  student  draw  a  figure  of  a  curve  carrying  it  to  the  half  circle  and  show 
why  the  180°  point  could  not  be  definitely  located  from  the  beginning. 


COMPOUND  CURVES. 


243 


FIG.  111. 


232.  Compound  curves.  These  are  used  in  railroad  and 
other  locations  to  bring  the  line  of  the  work  to  conform  more 
nearly  to  the  shape  of  the  surface  along  which  it  lies.  Fig.  Ill 
shows  such  a  curve  of  two  arcs 
only.  Compound  curves  have 
frequently  several  arcs  of  (iiffer- 
ent  radii.  In  compound  curves 
the  tangent  distances  are  always 
unequal,  and  in  the  case  of  two 
arcs  the  shorter  tangent  will  be 
adjacent  to  the  arc  of  shorter 
radius.  The  two  arcs  must  have 
a  common  tangent  at  their  junc- 
tion P.  The  sum  of  the  central 
angles  will,  of  course,  be  equal 
to  the  angle  A.  The  arcs  are 
laid  out  as  are  simple  curves,  one 
at  a  time.  The  arc  AP  is  laid  out  from  A  to  P,  and  the  tran- 
sit is  then  moved  to  P,  and  the  arc  PB  is  laid  out.  The 
problems  in  simple  and  compound  curves  are  merely  exercises 
in  the  Geometry  of  the  circle,  often  solved  by  Trigonometry. 

One  element  of  curves  is  of  much  assistance  in  solving  the 
problems  that  arise  in  laying  them  out  and  changing  them  to 
better  fit  the  ground  where  they  are  located.  This  element  is: 

PROPOSITION  :  If  two  circular  arcs  begin  at  a  common  tan- 
gent point  and  end  in  parallel  tangents,  their  long  chords  are 
coincident  in  direction  and  vary 
only  in  length. 

In  Fig.  112  let  J.Fand  VB 
be  the  tangents  to  the  curve  AB  ; 
and  AV  and  V C,  the  tangents 
to  the  curve  AC.  A  is  a  common 
tangent  point.  Since  AV  and 
A  V'  are  coincident,  and  VB  and 
V'O  are  parallel,  A  =  A'.  The 
deflection  angles  from  the  com- 
mon tangent  to  the  points  B  and 
O  will  each  equal  £A.  Therefore,  AB  is  coincident  with  A  0. 


Fro.  112. 


CHAPTER   IX. 

TOPOGRAPHICAL  SURVEYING. 
TOPOGRAPHY. 

233.  Methods  of  representing  surface  form.  A  topographical 
map  is  a  map  showing  the  conformation  of  the  surface  of  the 
area  of  land  mapped. 


•I  Scale.  1  in.=  1000  feet 


^//m'ni    "wv^W^ 
FIG.  113. 

A  topographical  survey  is  a  survey  conducted  for  the  pur- 
pose of  obtaining  information  for  the  production  of  a  topo- 
graphical map  of  the  area  surveyed. 

244 


TOPOGRAPHY. 


245 


There  are  two  common  ways  of  representing  the  conforma- 
tion of  any  small  portion  of  the  earth's  surface. 

(1)  By  means  of  hachures  or,  as  more  commonly  called, 
hill  shading  or  hatching.  Fig.  113  is  an  example  of  this 
method.  It  is  evident  that  it  is  little  better  than  sketching, 
since  it  gives  no  definite  notion  of  the  height  of  the  hills  or  the 
rate  of  slope  of  the  surface.  It  is  now  employed  only  for  the 


FIG.  114. 


purpose  of  making  pictures  to  be  intelligible  to  those  not  ac- 
quainted with  technical  methods  for  representing  surfaces,  and 
to  convey  to  them  as  well  as  to  trained  surveyors,  a  general 
notion  of  the  conformation  of  the  surface  where  no  definite 
information  is  required. 

(2)  By  means  of  contours,  or  lines  of  equal  elevation. 
Fig.  114  is  an  example  of  this  method. 

The  word  "  contour  "  as  used  in  surveying  means  a  line  all 


246 


TOPOGRAPHICAL   SURVEYING. 


points  of  which  have  the  same  elevation  above  or  below  some 
assumed  level  surface,  as  sea  level.  Let  the  student  imagine 
some  section  of  country  with  which  he  is  familiar  covered 
with  water  so  deep  that  the  tops  of  the  highest  hills  are 
just  in  the  surface  of  the  water.  Let  it  be  imagined  that 
the  water  flows  away  till  the  surface  has  dropped  five  feet  ver- 
tically. The  intersection  of  the  water  surface  and  the  land 
surface  would  be  a  five-foot  contour  line.  Let  the  surface  drop 
five  feet  more,  and  there  would  result  another  five-foot  contour 
line.  The  contour  line  may  then  be  said  to  be  a  level  line 
lying  wholly  in  the  surface  of  the  earth. 

The  contours  of  a  theoretically  perfect  conical  hill  would 
be  a  series  of  circles  having  their  centers  in  the  axis  of  the 
cone.  A  contour  map  of  such  a  hill  would  be  a  series  of  con- 
centric circles.  This  is  seen  from  Fig.  115.  If  it  is  assumed 

that  the  base  of  the  hill  has  an 
elevation  of  one  hundred  feet 
above  some  datum,  then  the 
contour  map  showing  "  ten-foot 
contours"  would  be  as  in  the 
lower  part  of  the  figure.  The 
horizontal  distance  between  any 
two  contours  shown  will  indi- 
cate the  rate  of  slope  of  the  sur- 
face. The  direction  of  steepest 
slope  is  always  in  the  direction 
of  the  common  perpendicular  to 
two  consecutive  contours.  If 
two  contours  of  the  same  eleva- 
tion were  to  be  found  on  a  map 
running  together  and  continu- 
ing as  a  single  line  and  then  separating  again,  there  would  be 
indicated  a  perfect  knife-edge  ridge  or  hollow.  Since  such  a 
thing  does  not  exist  in  nature,  no  such  representation  can  be 
found  on  a  correct  contour  map.  Moreover,  except  in  the  case 
of  an  overhanging  bluff  or  a  cave,  contours  of  the  same  or  of 
different  elevations  could  not  cross,  and  hence  a  correct  contour 
map  will  not  show  contour  lines  crossing.  A  contour  line  that 
closes  upon  itself  indicates  a  peak  higher  than  the  surrounding 


FIG.  115. 


TOPOGRAPHY. 


247 


ground  or  a  hole  without  an  outlet.  Since  any  contour  line, 
except  one  that  closes  on  itself  as  above  described,  must  con- 
tinue indefinitely,  closing  on  itself  somewhere,  to  be  sure,  no 
contour  line  can  begin  or  end  within  the  border  of  a  map 
unless  it  closes  on  itself.  With  the  exception  noted,  all  con- 
tours must  appear  first  in  the  border  of  the  map  and  disappear 
in  the  same  or  another  edge.  When  contours  cross  a  stream, 
it  is  customary  in  mapping  to  run  them  up  to  one  bank  and 
stop  them,  beginning  again  on  the  opposite  bank.  This  merely 
means  that  the  portions  that  run  up  the  banks  of  the  stream 
and  unite  in  the  bed,  are  omitted.  In  some  cases,  however, 
the  contours  are  carried  across  the  stream. 

A  contour  map  is  thus  seen  to  show  with  definiteness  the 
conformation  of  the  surface. 


234.  Field  methods  for  small  area.  There  are  various  ways 
of  making  topographical  surveys,  according  to  the  use  to  which 
the  information  obtained  is  to  be  put,  or  the  extent  of  terri- 
tory to  be  mapped.  A  more  or  less  precise  and  extensive 
topographical  survey  is  required  when  a  large  and  important 
building  is  to  be  erected,  when  a 
park  is  to  be  laid  out,  or  any  land- 
scape gardening  done,  when  a  rail- 
road, canal,  or  road  is  to  be  built, 
or  when  a  system  of  irrigation  or 
drainage  is  to  be  planned. 

When  the  area  to  be  mapped 
is  small,  as  a  city  lot,  or  block, 
or  a  very  small  farm,  the  follow- 
ing method  is  used  : 

The  area  is  divided  into  squares  or  rectangles  of  from 
twenty  feet  te  one  hundred  feet  on  a  side,  stakes  being  driven 
at  each  corner.  The  stakes  along  one  edge  will  usually  be 
lettered  and  along  another  edge  they  will  be  numbered.  Any 
interior  stake  would  then,  be  known  by  its  number  in  its 
lettered  row :  thus,  stake  X  (Fig.  116)  would  be  known  as 
stake  Dy  Levels  are  now  taken  at  all  corners  and  any  points 
of  abrupt  change  of  slope  not  at  a  corner.  From  the  level  notes 
the  contours  are  obtained,  as  described  below.  Contours  are 


FIG.  116. 


248  TOPOGRAPHICAL  SURVEYING. 

usually  named  with  their  elevations  above  the  assumed  datum. 
Thus,  there  would  be  the  110-foot  contour,  the  112-foot  con- 
tour, etc.  In  work  covering  small  areas  it  will  usually  be 
necessary  to  locate  contours  with  vertical  intervals  of  from  one 
foot  to  five  feet.  In  one  map  the  interval  between  contours  will 
be  uniform. 

235.  Contour  map.     To  draw  the   contour   map   from  the 
notes  that  have  been  taken,  lay  out  in  pencil,  on  paper,  to  scale, 
a  map  of  the  squares  that  have  been  used  in  the  field.     Note 
lightly  in  pencil  the  elevations  of  the  points  where  levels  have 
been  taken.     Assume  that  the  slope  along  any  line  is  uniform 
from   one  known  elevation  to  the  next.     If  the  contours  are 
being  drawn  with  an  interval  of  two  feet,  for  instance,  and  if 
either  of  the  known  elevations  is  an  even  two  feet,  as  112  feet 
or   114   feet,  then   a  contour  line  passes  through  the  known 
elevation  point.     If  not,  and  this  will  be  the  usual  case,  the 
point  where  the  contour  crosses  the  line  between  the  two  points 
of  known  elevation  must  be  found   by  interpolation.     When 
found,  the  point  is  numbered  with  the  elevation  of   the  line 
passing  through  it,  and  other  points  are  found.     This  is  done 
over  the  whole  map,  and  finally  points  of  equal  elevation  are 
connected,  giving  contour  lines. 

The  lines  will  not  usually  be  drawn  straight  between  points, 
but  will  be  curved,  either  from  a  knowledge  of  the  ground  or, 
in  the  absence  of  such  information,  so  as  to  form  no  angles,  but 
to  make  smooth,  curved  lines.  Sometimes  a  profile  of  each  line 
is  drawn  in  pencil,  as  indicated  for  line  6r,  and  on  this  profile 
are  noted  the  points  having  elevations  of  required  contours. 
These  are  projected  down  on  to  the  line,  giving  the  points  on 
the  line  intersected  by  the  given  contours.  The  contour  lines 
are  inked  in  in  black  or  brown  or  red,  according  to  the  purpose 
of  the  map.  The  outside  lines  of  the  square  are  inked  in  black. 
The  contour  lines  are  numbered  with  their  elevations  usually  at 
every  fifth  contour  or  at  the  even  tens. 

236.  Field  methods  for  large  area.     When  the  area  is  large, 
as  a  farm  of  twenty  or  more  acres,  or  even  less,  or  a  city  or 
county,  or  a  portion  of  a  given  valley,  the  following  method  is 
adopted : 


TOPOGRAPHY.  249 

Points  are  located,  more  or  less  at  random,  over  the  area  to 
be  mapped,  and  their  elevations  are  obtained.  The  points  are 
then  mapped,  and  it  is  assumed  that  the  slope  is  uniform 
between  those  that  are  adjacent ;  and  the  positions  of  contours 
between  points  are  determined  by  interpolation,  as  was  done 
with  the  squares.  Equal  elevations  are  then  connected  by 
curved  lines,  which  are  the  contour  lines.  Sketches  are  made 
in  the  field  to  assist  in  shaping  the  contours. 

237.  Transit  and  stadia  method.  In  the  field  the  points 
are  located  by  "polar  coordinates"  ;  that  is,  by  direction  and 
distance  from  a  known  point.  This  work,  as  well  as  the 
determination  of  elevations,  is  best  done  with  a  transit  and 
stadia. 

The  field  work  is  as  follows  :  A  starting  point,  as  the  inter- 
section of  two  streets,  the  corner  of  a  farm,  or  simply  an 
arbitrary  spot,  is  established  by  driving  a  firm  stake  that  will 
not  be  easily  disturbed.  The  transit  is  set  over  this  point  with 
vernier  reading  zero,  and  the  instrument  is  pointed,  by  the 
lower  motion,  in  the  direction  of  the  meridian.  This  may  be 
the  true  meridian  previously  determined,  the  magnetic  merid- 
ian as  shown  by  the  needle,  or  an  arbitrary  meridian  assumed 
for  the  purpose  of  the  survey.  If  either  of  the  latter,  a  stake 
is  driven  a  distance  away,  so  that  the  direction  may  be  again 
accurately  found.  It  is  assumed  that  the  true  meridian  will 
have  been  so  marked  as  to  be  found  again.  The  elevation  of 
the  point  over  which  the  instrument  is  set,  if  not  known, 
is  assumed  and  written  in  the  notebook.  A  traverse  line  is 
now  run,  noting  the  azimuths  of  all  courses,  as  described  in 
Chapter  V. 

The  distances  are  measured  with  the  stadia  and  recorded  in 
the  book.  The  vertical  angles  are  also  noted.  Vertical  angles 
and  distances  are  read  from  both  ends  of  any  course  as  a  check. 
The  traverse  line  is  chosen  with  a  view  to  obtaining  from  each 
station  the  largest  possible  number  of  pointings  to  salient 
features  of  the  area  to  be  surveyed,  and,  while  the  instrument 
is  set  at  any  station  and  before  the  traverse  line  is  completed, 
these  are  taken.  These  additional  pointings  are  usually 
called  "side  shots."  For  each  pointing  the  distance,  azimuth. 


250  TOPOGRAPHICAL  SURVEYING. 

and  vertical  angle  are  read.  These  readings  will  locate  the 
point  and  determine  its  elevation. 

After  the  instrument  has  been  oriented  at  any  station,  a 
rod  is  held  alongside  and  the  height  of  the  axis  of  the  tele- 
scope is  noted  in  the  book  as  a  heading  to  the  notes  taken 
from  that  station.  While  this  is  being  done,  another  rodman 
(there  are  usually  two  or  more)  selects,  by  the  direction  of  the 
person  in  charge  of  the  survey  or  by  his  own  judgment,  a 
second  point  to  be  occupied  by  the  transit.  He  drives  a  stake 
and  marks  it  HI  (Stadia  Station  1).  The  rodmen  are  provided 
with  hatchets  for  the  purpose  of  making  and  driving  stakes 
and  clearing  away  trifling  obstructions  to  sight.  If  there  is 
much  clearing  to  do,  axmen  are  employed. 

The  rodman  holds  his  rod  vertical  and  just  behind  the 
stake  from  the  instrument,  with  its  edge  turned  toward  the 
observer.  The  observer  directs,  by  the  upper  azimuth  motion, 
the  line  of  collimation  to  the  edge  of  the  rod,  making  the  exact 
bisection  by  the  tangent  screw.  He  then  reads  the  azimuth 
and  calls  it  to  the  recorder,  who  records  it.  He  also  reads  the 
needle,  and  this  is  likewise  recorded,  to  serve  as  a  witness 
should  an  error  be  discovered  in  mapping  the  work.  Having 
read  the  azimuth,  the  observer  signals  the  rodman,  who  turns 
the  graduated  side  of  the  rod  toward  the  observer.  The  latter 
turns  the  middle  horizontal  wire  to  a  point  on  the  rod  corre- 
sponding to  the  height  of  the  instrument,  using  the  clamp 
and  slow  motion  screw  of  the  vertical  motion.  He  then  notes 
the  lower  wire  to  see  whether  it  is  near  a  fifty  or  hundred 
graduation,  makes  it  coincide  with  such  graduation,  and  reads 
the  distance.  He  again  brings  the  horizontal  wire  to  the  proper 
height  and  signals  that  he  is  through.  He  then  reads  the 
vertical  angle,  which  is  recorded. 

While  this  is  being  done,  the  rodman  who  was  at  the  instru- 
ment selects  by  direction  or  judgment  a  point  for  a  side  shot,  and 
stands  there  with  the  graduated  side  of  the  rod  toward  the  ob-. 
server.  The  observer  now  unclamps  the  upper  motion,  and  does 
not,  as  a  rule,  clamp  it  again  till  he  leaves  the  station.  He 
turns  the  line  of  sight  on  the  rod,  and  after  reading  the  distance 
he  reads  the  azimuth  —  the  needle  is  neglected  on  side  shots  — 
and  vertical  angle  as  already  described  for  the  stadia  station, 


TOPOGRAPHY.  251 

and  signals  that  he  is  through.  The  rodman  who  set  the 
station  has  now  found  a  point,  and  his  rod  is  observed  as  last 
described,  and  the  work  continues  in  this  way,  observing  first  on 
one  rod  and  then  on  another,  according  to  which  is  first  ready. 

The  points  that  are  selected  for  side  shots  are  such  as  will 
enable  contours  to  be  well  drawn  on  a  map  on  which  those 
points  are  plotted.  They  should  be  therefore  along  ridges  and 
hollows  at  all  changes  of  the  slope.  They  should  be  taken 
frequently  along  a  stream  to  locate  its  course.  It  is  usually 
required  to  obtain  information  for  the  mapping  of  artificial 
structures,  as  roads,  fences,  houses,  etc.  Therefore  pointings 
should  be  taken  to  all  fence  corners  and  angles,  to  enough 
corners  of  all  buildings  to  enable  the  building  to  be  mapped, 
and  to  a  sufficient  number  of  points  in  all  roads.  If  it  is  de- 
sired to  indicate  wooded  lands,  then  pointings  must  be  taken 
around  the  edges  of  timber.  While  all  this  is  being  done,  the 
recorder  is  recording  the  notes  and  endeavoring  to  keep  a 
sketch  on  the  right-hand  page  of  the  book  to  assist  the  mem- 
ory in  mapping. 

After  all  the  side  shots  from  CDo  have  been  taken,  the  in- 
strument is  moved  to  E^  and  a  rodman  goes  to  QQ-  The 
instrument  is  oriented  on  E^,  the  edge  of  the  rod  being  first 
sighted.  The  needle  is  read  and  recorded,  as  well  as  the  dis- 
tance and  vertical  angle.  After  one  of  the  rodmen,  who  has 
come  up  to  the  station,  has  noted  the  height  of  the  instrument, 
he  goes  ahead  to  select  a  new  stadia  station,  which  is  estab- 
lished and  located  as  was  Qx,  and  the  work  proceeds.  Instead 
of  requiring  a  rodman  to  be  at  the  instrument  each  time  a 
new  point  is  occupied,  the  recorder  or  observer  may  carry  a 
light  graduated  rod  with  which  to  determine  the  height  of 
the  transit.  It  is  convenient  to  have  the  five-foot  point  of 
the  stadia  painted  red  to  expedite  the  setting  of  the  wire 
on  the  proper  point  for  the  determination  of  the  vertical 
angle. 

The  form  of  notes  used  in  stadia  surveying  is  given  on 
page  252.  The  left  page  of  the  notebook  is  represented.  The 
right  page  is  used  for  sketching  and  for  additional  notes  de- 
scriptive of  the  stations  occupied  or  observed.  The  sketch 
should  be  as  full  as  possible  and  neatly  drawn. 


252 


TOPOGRAPHICAL  SURVEYING. 


RENS.  POLY.  INST.  PRACTICE  SURVEY.    JUNE,  1894. 


INST.  ON  Q0.    ELEV. 


H.  I.  4.90.     SATURDAY,  14. 


ADEY,  OBSERVER. 
DAVIS,  RECORDER. 


STATION. 

AZIMUTH. 

DISTANCE. 

VERT. 
ANGLE. 

DlFF. 

ELEV. 

ELEV. 

331°  39' 

556 

-  1°  41' 

-  16.3 

W.  side  road  .... 
Beginning  slope  .  .  . 
C.  P.  (Contour  point)  . 
C  P 

34    30 
243    08 
31    21 

286    48 

61 

80 
119 
110 

-0    50 
-0    22 
+  1    05 
2    00 

-    0.9 
-    0.5 
+    2.2 
3  8 

829.1 
829.5 
832.2 
826  2 

c  p  

19    39 

141 

+  1    10 

+    2.9 

832.9 

Top  slope  

313    26 

45 

-  1    20 

-    1.0 

829.0 

C.  P.  west  road  .  .  . 
C.  P.  on  hill  .... 
C.  P  
C  P 

356    20 
347    32 
331    24 

334    05 

123 
94 
103 
120 

-3    00 
-3    47 
-5    20 
5    40 

-    6.4 
-    6.2 
-    9.4 
-  11  7 

823.6 
823.8 
820.6 
818  3 

INST.  ON 


ELEV.  813.7.    H.  I.  5.0. 


151  39 
352  05 
152  45 

556 
69 
436 

+  1    41 
+  2    55 
+  0    31 

+  16.3 
+    3.5 

+    3.9 

817.6 

Qj  2  (described  on  rt.  p.) 
C.  P.  on  hill     .... 

C.  P  

151  56 

289 

-1    00 

-    5.1 

808.6 

C.  P.,  W.  side  road  .     . 
C   P 

141  11 
151  16 
141  33 

191 
198 
305 

+  0    35 
-1    26 
+  0    35 

+    2.0 
-    4.9 
+    3.6 

815.7 
808.8 
817.3 

C.  P.,  E.  side  road     .     . 

Beginning  of  cross  rd.W. 

145  47 

97 

-0    55 

-    1.6 

812.1 

INST.  ON  H2.    ELEV.  817.2.    H.  I.  4.9. 

Hi    • 

172  05 
348  09 

69 
156 

-2    54 
-2    11 

-    3.5 
-    6.1 

etc. 

etc. 

The  elevations  of  the  stadia  stations  are  not  written  in  the 
sixth  column,  but  in  each  heading.  It  sometimes  happens  that 
the  vertical  angles  do  not  check  so  well  as  in  the  example  given. 
When  this  is  the  case,  the  mean  value  of  the  difference  of  ele- 
vation is  used  for  determining  the  elevation. 

An  instrument  that  has  poor  plate  bubbles  is  not  suitable 


SIMPLE    TRIANGULATION.  253 

for  stadia  work.     To  secure  good  work  with  such  an  instrument 
it  is  necessary  to  use  the  telescope  bubble  for  leveling  it. 

If  it  is  necessary  to  correct  the  horizontal  distance,  it  is 
done  by  the  use  of  the  table  in  Appendix,  page  380.  The  dif- 
ferences in  elevation  are  obtained  from  the  same  table,  a  dia- 
gram, or  Colby's  slide  rule. 

SIMPLE  TRIANGULATION. 

238.  When  used.     If  the  area  to  be  surveyed  covers  sev- 
eral square  miles,  or  is  an  elongated  strip,  as  a  narrow  valley, 
it  is  best  to  hang  the  topography  on  a  system  of  triangles. 
Unless  this  area  is  extensive,  as  a  large  county  or  a  state,  these 
triangles  may  be  considered  plane  triangles,  the  curvature  of 
the  earth  being  neglected.     In  case  the  triangles  are  to  be  con- 
sidered as  plane,  it  will  be  best  to  assume  for  the  meridian  of 
the  survey  a  true  meridian  near  the  middle  of  the  area,  east 
and  west.     The  process  of  laying  out  such  a  system  of  triangles 
is  as  follows  : 

239.  Measuring  the  base  line.     A   fairly  level  stretch  of 
ground  is  selected,  if  possible,  near  the  middle  of  the  area  to 
be  mapped,  and  a  line  of  from  one  thousand  feet  to  a  half 
mile  long,  or  longer,  is  measured  very  carefully.     This  line  is 
called  the  base  line.     Its  ends  are  marked  with  substantial 
monuments  of  stone  or  with  solid  stakes.     If  the  survey  is  of 
sufficient  importance  to  warrant  the  permanent  preservation  of 
the  system  of  triangles  for  future  reference,  the  monuments 
that  mark  the  base  line,  and  those  that  mark  the  apexes  of 
triangles,  should   be   stones  not  less  than  six   inches  square 
in  cross  section  and  two  feet  long.     In  the  top  face,  which 
should  be  dressed,  there  is  drilled  a  hole  large  enough  to  take 
a  half -inch  copper  bolt,  which  is  leaded  in.     The  head  of  the 
bolt  is  marked  writh  a  cross  to  determine  the  exact  point,  and 
the  stone  is  set  deep  enough  to  be  beyond  the  disturbing  action 
of  frost.     The  point  may  be  carried  to  the  surface  by  "  plumb- 
ing," and  a  temporary  point  set  for  use.     The  location  of  each 
monument  should  be  fully  described  with  reference  to  sur- 
rounding objects  of  a  permanent  character,  so  that  it  may  be 
readily  found  at  a  future  date.     If  no  permanent  objects  that 


254  TOPOGRAPHICAL   SURVEYING. 

can  be  used  are  to  be  found,  then  constructions  of  stone,  brick, 
or  timber  are  built  nearly  on  the  surface  of  the  ground,  and 
the  point  is  described  with  reference  to  these.  Such  precau- 
tions are  usually  necessary  only  for  extensive  geodetic  surveys. 

The  measurement  of  the  base  line  for  the  surveys  of  smaller 
areas,  which  are  the  only  surveys  to  be  described  here,  should 
be  made  with  a  precision  of  from  one  in  five  thousand  to  one 
in  fifty  thousand,  according  to  the  scale  of  the  map  that  is  to 
be  made,  the  area  surveyed,  and  the  importance  of  the  work. 

The  two  ends  of  the  base  line  being  determined  and  marked, 
the  transit  is  set  over  one  end,  and  a  line  of  stakes  ranged  out 
between  the  two  ends.  The  alignment  is  made  as  perfect  as 
possible,  and  the  stakes  are  set  as  nearly  as  practicable  at 
single  tape  lengths  apart,  center  to  center,  or  at  half  or  quar- 
ter tape  lengths.  These  stakes  are  made  quite  firm  in  the 
ground,  and  should  be  not  less  than  two  inches  square  in  sec- 
tion. For  high-class  work  the  stakes  should  have  on  their 
tops  tin  or  zinc  strips  on  which  to  mark  the  measurements 
with  fine  scratches. 

The  measurements  are  then  made,  the  temperature,  pull, 
and  grade  elements  being  noted,  and  the  distance  between  sup- 
ports. The  requisites  for  securing  the  required  degree  of  pre- 
cision are  mentioned  in  Chapter  I.  If  the  absolute  length  of 
the  tape  at  some  standard  temperature  and  pull  is  unknown, 
of  course  the  absolute  length  of  the  line  remains  undetermined. 
For  a  precision  of  one  in  one  million,  which  is  attempted  in  the 
measurement  of  base  lines  for  extensive  geodetic  surveys,  as  of 
a  state  or  continent,  much  more  refined  methods  than  those 
described  are  necessary.  These  methods  properly  belong  to 
the  subject  of  geodetic  surveying. 

240.  Measuring  the  angles.  After  the  base  line  is  estab- 
lished and  measured,  or  even  while  it  is  being  measured,  prom- 
inent points  in  the  area  are  chosen  as  "  triangulation  stations," 
that  is,  apexes  of  triangles.  These  are  so  chosen  that  no  angle 
shall  be  less  than  30  degrees,  nor  more  than  120  degrees.  The 
first  point  is  chosen,  as  nearly  as  possible,  in  a  line  at  right 
angles  to  the  base  line  at  its  middle  point.  A  second  point 
may  be  chosen  opposite  the  first  one,  and  the  others  are  then 


SIMPLE   TR1ANGULATION.  255 

located  with  reference  to  these  and  to  one  another.  When 
these  stations  are  established,  signals  are  erected  over  them. 
For  work  of  the  magnitude  here  considered,  these  may  be  simply 
wooden  poles,  about  one  and  one  half  inches  in  diameter,  to 
which  a  flag  of  some  white  and  dark  cloth  is  tacked  to  make 
the  signal  more  easily  found.  Such  signals  are  usually  called 
flags.  A  transit  is  then  set  over  one  extremity  of  the  base  line, 
and  the  angles  formed  at  that  point  by  lines  to  the  adjacent 
stations  are  measured.  When  all  of  these  angles  have  been 
measured,  the  transit  is  moved  to  the  other  extremity  of  the 
base  line,  and  similar  measurements  are  made  there,  and  subse- 
quently at  each  triangulation  station.  There  are  two  methods 
for  measuring  angles,  called  respectively  the  method  by  repeti- 
tion and  the  method  by  continuous  reading. 

In  the  first  method,  each  angle  is  measured  separately  a 
number  of  times,  and  the  mean  result  is  taken.  The  process  is 
as  follows  :  With  vernier  at  0°  OO',1  set  by  lower  motion  on  the 
left-hand  station. 

Unclamp  above,  set  on  right  station. 
Unclamp  below,  set  on  left  station. 
Unclamp  above,  set  on  right  station. 

Continue  as  many  times  as  may  be  necessary  to  use  practically 
the  whole  circle  of  360°,  and  finally  read.  Divide  the  reading 
by  the  number  of  repetitions  for  the  value  of  the  angle. 
Reverse  the  telescope  and  repeat  the  above  operation  from 
right  to  left.  The  readings  are  taken  in  both  directions  to 
eliminate  errors  due  to  clamping  and  unclamping  and  personal 
mistakes  in  setting.  They  are  taken  with  the  telescope  both 
direct  and  reverse,  to  eliminate  errors  of  adjustment.  They 
are  made  to  include  the  whole  circle  to  eliminate  errors  of 
graduation.  Both  verniers  are  usually  read  to  eliminate  the 
effect  of  eccentricity  of  verniers. 

When  such  an  instrument  as  the  engineer's  transit  is  used, 
this  method  is  the  simplest  and  probably  the  best. 

The  second  method,  by  continuous  reading,  differs  from 
the  first  in  that  each  angle  is  not  measured  independently  of 

1  Any  other  setting  could  be  used,  tut  it  is  considered  better  in  this  method  to 
set  at  zero. 


256  TOPOGRAPHICAL   SURVEYING. 

the  others  at  the  point  occupied.  The  telescope  is  pointed  at 
each  of  the  distant  stations  consecutively,  and  the  vernier  is 
read  for  each  pointing,  —  the  difference  between  the  consecu- 
tive readings  being  the  angle  between  the  corresponding  points. 
Thus,  in  Fig.  117,  the  instrument  being  at  /,  the  telescope 
is  first  pointed  to  A  and  the  vernier 

A^  P  is  read,  then  to  B  and  the  vernier  is 

read,  then  to  (7,  D,  E,  F,  and  A  in 
succession.     The   reading   on  A  sub- 

F — -~---J\j£_____----- — c  tracted  from  the  reading  on  B  gives 
the  angle  AIB.  In  this  method  it  is 
necessary  to  read  both  to  the  right 
and  left,  and  with  the  telescope  both 
direct  and  inverted.  Moreover,  since 

H'lG.   117.  i  ^         •  i  T 

each  angle  is   measured   on  only  one 

part  of  the  limb,  it  is  necessary,  after  completing  the  readings 
once  around  and  back,  to  shift  the  vernier  to  another  part 
of  the  limb  and  repeat  the  readings  both  forward  and  back, 
with  telescope  both  direct  and  inverted.  This  is  done  as 
many  times  as  there  are  sets  of  readings.  Each  set  of  com- 
plete readings  to  right  and  left,  with  telescope  direct  and 
inverted,  gives  one  value  for  each  angle.  If  it  is  desired  to 
measure  each  angle  three  or  five  times,  then  three  or  five  sets 

of  readings  must  be  taken.     The  amount  that  the  vernier  is 

-i  ono 
shifted  between  each  two  sets  is ,  where  n  is  the  number  of 

sets.1 

241.  Notes.  A  good  form  for  keeping  the  record  of  the 
notes  of  the  triangulation,  assuming  that  it  is  done  by  the  re- 
peating method  and  with  an  engineer's  transit  reading  to  thirty 
seconds,  is  shown  on  page  257. 

The  notes  of  one  set  of  readings  are  given.  As  many 
sets  as  may  be  desired  are  taken.  For  the  purposes  of  a  small 
topographical  survey,  one,  or  at  most  two,  sets  will  be  sufficient. 
The  check  marks  indicate  the  number  of  times  the  angle  is 
measured,  a  check  being  made  after  each  measurement.  Three 
hundred  and  sixty  degrees  must  be  added  to  the  reading  to  the 
right  each  time  the  vernier  passes  the  zero  of  the  limb,  or  must 

l  The  student  should  make  a  programme  for  observations  by  this  method. 


SIMPLE   TRIANGULATION. 


257 


SATURDAY,  JUNE  24,  1894.     A.M. 

INSTRUMENT  ON  A1  A. 


STATION 
OBSERVED. 

VERNIER  A 

B 

MEAN  READING. 

COMPUTATION 

AND 

MEAN  ANGLE. 

A  B  .  .  .  . 
Inst.  direct  .  . 
AC.... 

0°  00'  00" 

360 
15°  02'  00" 

-30" 

or  30" 

359°  59'  45" 
15°  Or  45" 

15  01  45 
360 
360 
735  01  45 
359  59  45 

6)375  02  00 

62°  30'  20" 

AC.... 
Inst.  reversed  . 
A  B  .... 

15°  02'  00" 

V  V  \/  V  V  v' 
360 
360 

359°  58'  30" 

01'  30" 
58'  00" 

15°  01'  45" 
359°  58'  15" 

15  01  45 
360 
360 

735  01  45 
359  58  15 

6)375  03  30 

62°  30'  35" 

be  subtracted  from  the  readings  to  the  left.  The  latter  process 
brings  the  same  result  as  adding  to  the  initial  reading. 

If  the  method  of  continuous  reading  is  used,  the  form  of 
record  shown  on  page  258  is  considered  good. 

The  readings  are  begun  on  the  line  AB  of  the  triangu- 
lation  system,  and  carried  to  the  right  and  then  back,  the 
readings  taken  to  the  right  being  recorded  down  the  page,  and 
the  reverse  readings  up  the  page.  Such  a  series,  when  complete, 
forms  a  set  from  which  a  value  of  each  angle  is  determined. 
Provision  is  made  in  the  form  given,  for  three  sets  of  readings, 
while  but  one  complete  set  is  shown.  The  mean  of  the  three 
determinations  of  each  angle  would  be  taken.  The  errors  are 
purposely  shown  large.  With  a  transit  reading  to  thirty  sec- 
onds, there  will  usually  be  no  appreciable  difference  in  vernier 
readings,  and  it  is  always  possible  with  a  little  practice  to  read 
the  vernier  by  estimation  to  one  half  of  its  least  count. 

A  third  method  of  measuring  angles,  which  may  be  called 
a  combination  of  the  first  two,  saves  some  time,  and  gives 
results  as  good  as  the  class  of  work  warrants.  It  consists  in 

1  Triangulation  station. 
R'M'D  SDRV.  — 17 


258 


TOPOGRAPHICAL  SURVEYING. 


SATURDAY,  JUNE  24,  1894.    A.M. 

INSTRUMENT  ON  A  A. 


TELESCOPE. 

STATION 
OBSERVED 

VERNIER  A 

B 

MEAN  READING 

FROM  AB. 

MEAN  ANGLE 

WITH   AB. 

MEAN  ANGLE. 

Direct  .  . 
Inverted  . 

1 

000°  00'  00" 
359  59  30 

—  00'  30" 
59  00 

-000°  00'  15" 
45" 

-000°  00'  30" 

B   2 

3 

Direct  .  . 
Inverted  . 

1 

62  30  30 
62  30  00 

30  00 
29  30 

62  30  15 
62  29  45 

62  30  30 

62°  30'  30" 

C   2 

3 

1 

111  06  30 
111  06  30 

06  00 
0600 

111  06  15 
111  06  15 

111  06  45 

48  36  15 

D  2 

3 

1 

183  48  00 
183  48  00 

47  30 
47  30 

183  47  45 
183  47  45 

183  48  15 

72  41  30 

E   2 

3 

1 

278  12  00 
278  12  30 

11  30 
12  00 

278  11  45 
278  12  15 

278  12  30 

94  24  15 

F    2 

3 

•I 

359  59  30 
359  59  30 

5900 
5900 

359  59  15 
359  59  15 

359  59  15 

81  46  45 
359  59  15 

B   2 

3 

measuring  each  angle  a  stated  number  of  times,  reading,  meas- 
uring the  succeeding  angle  the  same  number  of  times,  reading, 
and  continuing.  After  all  the  angles  are  read  successively  to 
the  right,  the  telescope  is  inverted  and  pointed  on  the  right- 


SIMPLE   TRIANGULATION. 


259 


hand  station,  and  the  operation  is  repeated  to  the  left.  Three 
repetitions  of  each  angle  are  considered  sufficient.  This  pro- 
vides for  all  errors  except  those  of  graduation,  which  are 
usually  too  small  to  affect  work  done  to  the  degree  of  pre- 
cision required. 

The  form  of  notes  to  be  taken  with  this  third  method  is  : 

SATURDAY,  JUNE  24,  1894. 

INSTRUMENT  ON  A  A. 


i 

READINGS. 

MEAN 

i 

ME  AS. 

ANGLE 

WITH  AB. 

MEAN  ANGLB. 

VERNIER  ,4. 

B. 

MEAN. 

•  •• 

000°  00'  00" 

-1-00'  30" 

15" 

B 

000  00  30 

000°  00'  30" 

/w 

000  00  30 

01  00 

45 

C 

VVv/ 

187   31  00 

31  30 

15 
187  31  30 

187  31  30 

3)187°  31'  00" 

•w 

187  31  30 

32  00 

45 

62  30  20 

D 

"*" 

398  51  00 
398  51  30 

51  30 
52  00 

15 
398  51  30 
45 

398  51  30 

3)211   20  00 
70  26  40 

E 

vVv' 

567  59  30 

00  00 

45 
567   59  45 

567  59  45 

3)169  08  15 

v'vV 

567  59  30 

00  00 

45 

56  22  45 

242.  Adjusting  the  triangles.  All  the  angles  of  a  given 
triangle  are  measured.  If  but  two  were  measured  and  the 
third  computed,  the  entire  error  of  measurement  of  the  two 
angles  would  be  thrown  into  the  third  angle.  It  will  be  found 
almost  invariably,  on  adding  the  measured  angles  of  a  triangle, 
that  the  sum  of  the  three  angles  is  less  or  more  than  180°.  The 
error  should  be  less  than  one  minute,  using  the  engineer's  tran- 
sit. If  there  is  no  reason  to  suppose  that  one  angle  is  measured 
more  carefully  than  another,  this  error  is  divided  equally  among 
the  three  angles  of  the  triangle,  and  it  is  the  corrected  angles 
that  are  used  in  computing  the  azimuths  and  the  lengths  of 
the  sides.  This  distribution  of  the  error  is  called  "adjusting" 


260 


TOPOGRAPHICAL   SURVEYING. 


the  triangle.  With  the  large  systems  of  extensive  geodetic  sur- 
veys, very  much  more  elaborate  methods  are  employed,  since 
a  large  number  of  triangles  must  be  adjusted  simultaneously 
so  that  they  will  all  be  geometrically  consistent,  not  only  each 
by  itself,  but  one  with  another. 

243.  Computing  the  triangles.  When  all  the  angles  are 
measured  and  adjusted,  and  the  azimuth  of  the  base  line,  or 
any  other  line  of  the  system,  is  determined,  the  azimuth  of 
each  of  the  sides  may  be  found  by  computation.  It  is  to  be 
remembered  that  the  azimuth  of  any  line  xy  is  the  azimuth 
of  the  line  yx  +  180°.  From  the  measured  base  line  and  angles, 
the  lengths  of  each  side  of  the  system  may  be  computed  by  the 
ordinary  sine  formula. 

The  computations  can  be  much  facilitated  if  systematically 
arranged.  The  following  arrangement  of  the  computations 

of  the  system  of  triangles 
shown  in  Fig.  118  is  sug- 
gested. AB  is  supposed 
known.  The  adjusted 
angles  are  used.  AC  is 
determined  before  BC, 
because  BCis  to  be  used 
in  the  triangle  BCD. 

log  ,4 £- log  sinC,  =  log  AB 


FIG.  118. 


\ogAB     . 

log  sin  Cl 
.       AB 


log  sin  B} 

log  sin  A  .     .     . 

log  A  C  (value). 


log  sin  D 

.       BC 


log  sin  C2 

log  sin  B2. 
Etc. 


log 


AB 


+  log  sin  BI  =  log  A  C. 


log 


AB 

sinCl 


+  log  sin  A  =  log  -BC,  etc. 


Cologs  can  be  used  if  preferred. 
Write  the  values  found  for  A  C, 
BC,  etc.,  in  the  brackets. 

log  .         may  be  written  between 
sin  Cl 

log  sin  .Bj  and   log  sin  A  if    pre- 
ferred. 


244.   Use  of  triangles.     The  elevation  of  each  triangulation 
station  is  determined  by  leveling  from  some  bench  mark.     Any 


MAPPING.  261 

station  may  now  serve  as  a  beginning  point  for  the  topograph- 
ical survey.  The  transit  being  set  over  Station  X  is  oriented 
on  Station  Y  by  bringing  the  vernier  to  read  the  azimuth  of 
the  line  XY,  and  setting  the  line  of  collimation  on  Y  by  the 
lower  motion.  The  stadia  traverse  and  work  then  begin.  The 
azimuth  work  may  always  be  checked  in  the  field  by  occupying 
each  triangulation  station  as  it  is  reached,  and  determining  the 
azimuth  of  one  of  the  triangle  sides  radiating  from  the  station 
occupied.  If  this  agrees  with  the  known  azimuth,  the  work  is 
correct.  If  it  does  not  agree  within  one  or  two  minutes,  the 
work  should  be  rerun. 

The  elevations  also  serve  to  check  the  vertical  angles. 


MAPPING. 

245.  The  triangles.     If  the  topographical  survey  has  been 
based  on  a  system  of  triangles,  these  triangles  are  first  drawn. 
To  draw  them,  assume  one  station  as  the  origin  of  the  survey, 
and,  knowing  the  lengths  and  azimuths  of  all  the  sides,  compute 
the  latitude  and  longitude  of  each  station  with  reference  to  the 
assumed  origin.     Proceed  then  to  plot  the  stations  by  latitudes 
and  longitudes.     It  will  be  usual  to  assume  one  end  of  the  base 
line  as  the  origin,  and  the  direction  of  the  base  line  as  that  of 
the  reference  meridian.     If,  however,  the  work  is  referred  to 
another  meridian,  that  should  be  chosen  as  the  meridian  of 
reference  for  the  computing  and  plotting. 

The  lines  may  also  be  drawn  by  means  of  a  protractor  ;  but 
this,  while  quicker  and  sufficiently  precise,  if  a  large  protractor 
is  used,  for  much  work,  is  of  course  not  so  accurate  as  latitudes 
and  longitudes. 

Another  method  is  to  use  the  dividers  and  the  lengths  of 
the  triangle  sides,  describing  arcs  whose  intersections  locate 
the  triangulation  stations. 

246.  Outline  of  method  for  topography.    A  stadia  line  is  first 
plotted  to  see  that  it  closes  properly.     When  this  has  been 
satisfactorily  done,  the  side  shots  are  plotted.     Then  the  con- 
tours are  drawn  and  all  other  objects  that  it  is  desired  to  map. 
If  preferred,  of  course  the  detail  work  can  begin  before  the 


262 


TOPOGRAPHICAL   SURVEYING. 


completion  of  the  mapping  of  the  side  shots,  and  it  is  perhaps 
better  to  draw  in  buildings  and  other  structures  as  soon  as  the 
points  locating  them  are  plotted. 

247.  The  stadia  line.  The  best  method  for  plotting  the 
stadia  lines  is  by  latitudes  and  longitudes,  the  computations  for 
which  are  made  with  sufficient  exactness  by  a  diagram  or  by 
the  trigonometer  shown  in  Fig.  119.  The  arm  is  set  to  the 


Fm.  119. 

proper  azimuth  and  the  length  of  sight  is  noted  on  the  arm. 
The  latitude  and  longitude  differences  are  then  read  on  the 
cross-section  scale.1 

If  the  latitudes  and  longitudes  of  the  triangulation  stations 
have  been  determined,  it  will  be  known  on  the  completion  of 
computations  for  any  one  stadia  line  whether  or  not  that  line 
is  correct.  If  it  does  not  close  on  the  proper  triangulation 
station,  and  the  error  is  small,  that  error  may  be  distributed 

*  The  degree  numbering  should  be  continued  on  this  figure  to  360°,  and  the  signs 
for  latitude  and  longitude  differences,  marked  on  each  quadrant  of  numbers.  Let 
the  student  show  how  this  numbering  should  be  placed,  and  write  the  proper  signs. 


MAPPING. 


263 


among  the  various  courses,  as  in  a  land  survey.  If  the  error  is 
great,  it  must  be  found.  If  the  azimuths  were  checked  in  the 
field,  the  error  must  be  in  distance.  Note  the  direction  of  the 
closing  line  and  see  whether  it  corresponds  to  any  course.  If 
it  does,  the  error  is  probably  in  that  course,  and  it  should  be  re- 
measured.  If  it  does  not  correspond  to  any  one  course,  there 
is  probably  more  than  one  error.  In  case  of  error  of  geograph- 
ical position  due  to  mistakes  in  measurement,  the  elevations 
will  not  check  unless  the  error  is  made  on  practically  level 
ground;  hence  if  no  field  check  of  azimuths  has  been  possible 
and  the  elevations  check  while  geographical  posi- 
tions do  not,  the  error  is  in  azimuth. 


KEurrtLttSSEH.NV 


FIG.  120. 


The  stadia  line  may 
also  be  drawn  by  the 
use  of  a  protractor. 
The  protractor  should 
be  large,  either  one  of 
the  expensive  vernier 
protractors,  like  that 
shown  in  Fig.  120,  or 
a  paper  protractor, 
such  as  may  be  had 
from  any  dealer  in 
surveyors'  supplies.  The  paper  protractor  should  be  about 
fourteen  inches  in  diameter  and  graduated  to  quarter  degrees. 

248.  Side  shots.  It  may  be  prepared  for  use  in  mapping 
side  shots  as  follows,  see  Fig.  121  : l 

Number  the  graduations  from  zero  to  the  right  360°.  Draw 
through  the  zero  and  the  180°  points  and  the  center  a  straight 
line,  and  continue  the  line  to  the  end  of  the  sheet.  The  edges 
are  trimmed  true  and  square,  and  not  less  than  an  inch  of 
paper  is  left  outside  the  printed  circle.  With  a  pair  of  dividers 

i  Modification  of  a  form  suggested  by  Prof.  J.  B.  Johnson. 


264 


TOPOGRAPHICAL   SURVEYING. 


set  to  the  length  of  the  longest  average  shots,  when  measured 
to  the  scale  of  the  map,  describe  an  arc  acb  with  the  center  of 
the  protractor  as  a  center.  With  a  radius  equal  to  the  radius 
of  the  inner  circle  of  the  graduations,  and  with  a  center  on  the 
zero-1800  line  a  little  nearer  the  center  than  the  inner  circle  of 
the  graduations,  describe  an  arc  adb.  With  a  sharp  knife  cut 
out  the  crescent  thus  formed.  Cut  out  also  a  small  triangle, 
the  middle  of  one  side  of  which  shall  be  the  center  of  the  cir- 


FIG.  121. 

cie.  Midway  between  d  and  /  erect  a  perpendicular  to  dl,  as 
ef.  With  the  knife,  cut  along  the  lines  ea  and  fb.  Fold  the 
flap  eabf  thus  made  on  the  line  ef.  A  protractor  with  so  much 
material  taken  from  its  center  is  good  only  for  mapping  side 
shots. 

Perhaps  a  better  way  to  prepare  the  protractor  for  side 
shots  is  to  cut  out  about  three  quarters  of  the  circle  acb,  Fig. 
121,  leaving  a  trifle  more  than  a  quadrant  which  will  contain 
the  center  and  which  may  be  folded  back  after  the  protractor 


MAPPING.  265 

is  oriented  on  a  point.  This  form  will  give  a  somewhat  larger 
opening  for  plotting  the  shots.  If  a  field  map  is  to  be  made 
from  which  a  finished  map  is  to  be  traced,  it  may  be  made  on 
the  protractor  sheet,  or  larger  sheets  may  be  had  with  a  pro- 
tractor printed  on  them.  The  azimuths  may  be  transferred 
over  the  sheet  with  a  parallel  ruler. 

To  use  the  protractor,  draw  a  meridian  through  each  stadia 
station.  With  the  flap  unfolded,  center  and  orient  the  protrac- 
tor, weight  it  down,  and  fold  back  the  flap.  Paste  under  the 
zero  of  the  scale  that  is  to  be  used,  a  small  piece  of  drawing 
paper,  or  other  firm  paper,  through  which,  at  the  zero  gradu- 
ation, a  fine  needle  is  passed.  The  needle  is  stuck  into  the 
drawing  at  the  station,  and  the  rule  may  then  be  revolved  about 
it  as  a  center.  The  edge  of  the  scale  is  brought  to  each  azimuth 
in  succession,  and  the  distance  is  scaled  at  once  and  marked  on 
the  paper.  If  the  point  is  merely  taken  for  the  purpose  of 
obtaining  elevations,  its  elevation  is  noted  on  the  map  when  it 
is  plotted.  Otherwise,  the  name  of  the  object  is  marked  or  the 
object  is  sketched  in.  In  case  the  distance  is  so  great  that  the 
point  falls  under  the  paper  of  the  protractor,  a  mark  is  made 
with  the  name  or  elevation  of  the  point  and,  when  all  the  other 
points  are  plotted,  the  protractor  is  lifted  and  this  point  put 
in.  With  the  aid  of  the  sketches  and  the  assumption  that 
the  slope  is  regular  between  adjacent  plotted  points,  the  con- 
tours and  other  objects  are  now  drawn  in.  The  triangulation 
stations,  if  there  are  any,  should  be  inked  in  before  any  stadia 
plotting  is  done.  The  stadia  lines  are  not  drawn  nor  are  the 
stadia  stations  inked.  Buildings,  fence  lines,  etc.,  should  be 
inked  before  the  contours  are  drawn,  to  prevent  confusion.  It 
is  not  customary,  except  in  maps  of  small  scale,  to  run  contours 
across  a  road,  or  stream,  or  through  a  building. 

249.  Colby's  protractor.  A  very  convenient  form  of  pro- 
tractor for  this  work  is  Colby's,  shown  in  Fig.  122.  It  is  made 
in  three  pieces.  The  largest  piece  is  the  limb  L,  L,  Z,  gradu- 
ated from  0°  to  360°,  with  fifteen-minute  divisions  ;  E,  E,  are 
projections,  upon  which  weights  are  placed.  The  limb  has 
four  indexes  B,  B,  B,  B,  90°  apart,  by  which  the  protractor  is 
oriented.  A,  A,  A,  is  the  alidade,  fitted  to  revolve  inside  the 


266 


TOPOGRAPHICAL   SURVEYING. 


limb.  (7,  (7,  are  indexes  carried  by  the  alidade,  180°  apart. 
To  the  bar  D,  of  the  alidade  is  attached  the  scale  $,  by  the 
small  screws  and  nuts  N,  N.  The  scale  has  its  zero  mark  in 


FIG.  122. 


the  middle,  and  is  graduated  both  ways.  It  can  be  taken  off 
by  unscrewing  the  nuts  N,  N,  and  a  scale  of  different  denomi- 
nation may  be  substituted  when  desired. 

250.  Ockerson's  protractor.  Another  cheaper  form  is  shown 
in  Fig.  123.  It  is  centered  with  a  needle  point,  about  which 
it  will  revolve.  A  meridian  is  drawn  through  the  point  over 
which  the  protractor  is  to  be  set,  and  the  graduations  are  such 
that,  if  a  given  azimuth  is  brought  to  the  meridian  line,  the 
diameter,  on  which  is  a  scale  for  plotting  the  points,  is  in  the 
given  azimuth.  The  protractor  was  designed  by  Mr.  J.  A. 
Ockerson  for  use  in  the  Mississippi  River  surveys. 


MAPPING. 


267 


FIG.  123. 


251.  Finishing  the  map.     If   the  map  is  to  be  finished   in 
black,  it  may  be  finished  as  shown  on  Plate  I.,  at  the  end  of  the 
book.     If  it  is  to  be  finished  in  color,  a  very  effective  result, 
will  be  obtained  by  following  the  example  given  in  Plate  II., 
which  shows  the  scheme  in  use  by  the  United  States  Geological 
Survey.     Plate  III.  shows  a  more  effective  and  more  elaborate 
scheme.     The  contours  are  brown,  being  a  mixture  of  crimson 
lake  and  burnt  sienna.    The  red  is  vermilion.     Streams  and  all 
water  will  be  blue.   Roads  may  be  black  and  dotted  or  full  lines.1 

252.  Requirements  for  maps.     It  must  be  remembered  that 
no  map  is  complete  without  the  following  items  : 

(1)  A  neat,  explicit  title,  preferably  in  Roman  letters. 

(2)  Scale,  both  drawn  on  the  map  and  given  by  figures. 

(3)  The  date  of  the  survey. 

(4)  Name  of  surveyor  and  draughtsman. 

(5)  Direction  of  meridian. 

(6)  A  key  to  any  topographic  symbols  that  are  used. 

(7)  A  neat  line  border.2 

*An  excellent  work  on  "Topographical  Drawing  and  Surveying"  is  that  by 
Lieutenant  Henry  S.  Reed,  U.S.A. 

2  Other  requirements  for  land  maps  will  be  found  in  Appendix,  page  355. 


268  TOPOGRAPHICAL   SURVEYING. 

If  contours  are  drawn,  these  should  be  numbered,  to  make 
the  map  intelligible.  It  is  necessary  to  number  only  every 
fifth  or  tenth  line,  which  should  also  be  drawn  a  little  heavier 
than  the  others. 

253.  Scale.     The  scale  of  the  map  will  depend  on  the  terri- 
tory covered  and  the  use  to  which  the  map  is  to  be  put.     A 
survey  of  a  city  lot  may  be  mapped  on  a  scale  of,  say,  fifty  feet 
per  inch.     The  topographical  maps  of  the  United  States  Geo- 
logical Survey  are  on  a  scale  of  about  one  mile  per  inch.     A 
railroad  preliminary  survey  may  be  mapped  on  a  scale  of  from 
one  thousand  feet  per  inch,  for  ordinary  purposes,  to  one  hun- 
dred feet  per  inch,  for  close  detailing.     The  maps  of  the  United 
States  Coast  and  Geodetic  Survey  and  of   the  United  States 
Geological  Survey  are  mapped  to  a  "natural  scale,"  that  is, 
a  given  distance  on  the  map  is  some  round  fraction  of  the  dis- 
tance it  represents,  as  g^oiy  or  TtfthnF'  e^c>     Maps  of  land  sur- 
veys, or  others  made  to  show  to  non-technical  persons,  should 
be  made  at  so  many  feet,  or  chains,  or  miles,  to  the  inch ;  and 
the  top  of  the  map  should  be  north.     Maps  of  large  territories, 
mainly  for  the  use  of  those  who  are  conversant  with  surveying 
methods,  may  be  made  with  advantage  to  the  natural  scale. 
The  original  map  may  be  copied  by  tracing  on   cloth  or  on 
paper,  and  transferring  to  other  drawing  paper. 

THE  PLANE  TABLE. 

254.  Description.     Detail   topographical  surveys  are  some- 
times made  with  the  plane  table  ,  indeed,  this  is  the  standard 
instrument  of  the  United  States  Coast  Survey  for  such  work, 
and  is  also  largely  used  by  the  United  States  Geological  Survey. 

The  plane  table  consists  essentially  of  a  drawing  board 
with  a  suitable  leveling  device,  mounted  on  a  tripod,  and  a 
ruler  for  drawing.  The  ruler  is  attached  to  a  line  of  sight, 
usually  telescopic,  but  sometimes  merely  open  sights,  like  those 
of  the  compass.  The  combined  ruler  and  telescope  is  called 
the  alidade.  Fig.  124  is  a  complete  plane  table.  Clips  for 
holding  the  paper  are  shown,  as  well  as  the  alidade,  compass 
box,  and  levels,  and  a  device  for  setting  a  point  on  the  paper 
over  a  point  on  the  ground.  The  plane  table  is,  in  its  ordinary 


THE  PLANE  TABLE.  269 

form,  a  very  awkward  instrument,  and  is  used  very  little  out- 
side the  two  surveys  mentioned.  A  much  simpler  form  of 
leveling  head  than  that  generally  used  is  one  invented  by  Mr. 


FIG.  124. 


W.  D.  Johnson,  and  this  device  has  been  approved  by  the 
topographers  of  the  United  States  Geological  Survey.  It  is 
shown  in  Fig.  125.  By  loosening  the  wing  nut  d,  the  table 
may  be  leveled,  and  when  leveled,  the  nut  d  is  tightened. 

When  the  nut  g  is  loosened, 
the  table  may  be  turned  in  azi- 
muth without  disturbing  its  hori- 
zontality.  The  whole  arrange- 
ment is  very  light.  It  does  not 
permit  of  as  close  leveling  as 
does  the  ordinary  form  with 

leveling  screws,  and  should  not 

FIG.  125. 
be  used  where   contours   are  to 

be  carefully  determined  by  vertical  angles.  In  making  a  map 
of  a  park  showing  the  location  of  each  tree,  bush,  etc.,  for 
planning  new  work,  and  when  work  may  be,  without  loss, 
performed  on  pleasant  days  and  omitted  on  wet  or  damp  days, 


270  TOPOGRAPHICAL  SURVEYING. 

the  plane  table  may  be  used  with  advantage,  and,  aside  from 
the  United  States  Coast  and  Geological  Surveys,  it  is  for  this 
class  of  work  that  it  is  most  used.  The  author  prefers  for 
contour  work  to  use  the  transit  and  stadia.  If  the  advantage 
of  mapping  at  once  in  the  field  is  desired,  it  may  be  obtained 
by  having  an  assistant  and  a  light  drawing  board.  It  is  be- 
lieved that  a  lack  of  notes  is  rather  a  disadvantage  than,  as 
is  generally  assumed,  an  advantage. 

255.  Use.  The  plane  table  is  used  for  the  immediate  map- 
ping of  a  survey  made  with  it,  no  notes  of  angles  being  taken, 
but  the  lines  being  plotted  at  once  on  the  paper.  The  simplest 
case  is  the  location  of  a  number  of  places  from  one  point  by 
azimuth  and  distance.  The  table  is  set  up  so  that  some  con- 
venient point  on  the  paper  is  over  a  selected  spot  on  the  ground, 
and  clamped  in  azimuth.  The  ruler  is  then  brought  to  the 
point  on  the  paper  and  swung  about  it  till  the  line  of  sight 
which  is  parallel  in  azimuth  to  the  ruler1  is  directed  toward 
a  point  that  is  to  be  located.  A  scale  for  the  drawing  is  deter- 
mined, and  a  line  is  drawn  along  the  ruler  and  made  to  scale, 
equal  to  the  distance  to  the  desired  point,  which  distance  is 
found  by  measurement  or  by  the  stadia.  The  point  is  then 
located.  A  similar  procedure  locates  other  points.  This  is 
called  the  method  of  radiation.  If  a.  plane  table  is  set  up  in 
the  interior  of  a  field  all  of  whose  corners  are  visible  from  the 
position  of  the  table,  the  corners  may  be  thus  located  and  con- 
nected and  a  map  of  the  field  is  at  once  obtained.  There  is 
evidently  nothing  by  which  to  determine  the  area  except  to 
scale  the  map  for  additional  data  or  to  use  a  planimeter. 

Traversing  may  be  performed  with  the  plane  table  as  fol- 
lows :  If  it  is  a  field  that  is  to  be  "  run  out,"  set  the  table  over 
one  corner,  choosing  a  point  on  the  paper  to  represent  that 
corner  in  such  position  that  the  drawing  of  the  field  to  the  scale 
selected  will  come  on  the  board.  In  the  four-sided  field  shown 
in  Fig.  126,  it  would  be  necessary  to  occupy  but  two  corners 
to  map  the  field,  though  it  would  be  better  for  a  check  to 
occupy  the  third.  The  figure  shows  the  field  and,  to  a  very 
exaggerated  scale,  the  plane  table. 

1  It  is  only  necessary  that  the  two  shall  have  a  fixed  angle  in  azimuth. 


THE   PLANE   TABLE. 


271 


The  table  is  first  set  over  B  and  a  point  b  marked  on  the 
paper  directly  over  B.  The  instrument  being  clamped,  the 
alidade  is  brought  to  bear  on  A,  and  a  line  is  drawn  to  scale 
equal  to  BA,  which  is  found  by  measurement  or  by  stadia. 
The  alidade  is  then  directed  to  C,  and  the  line  be  is  drawn 
to  scale.  The  table  is  then  removed  to  C,  and  so  set  up  that 
the  point  c  shall  be  over  the  point  C  and  the  line  cb  in  the 
direction  CB.  This  is  hard  to  do  with  the  plane  table.  If 


s 

a. 

FIG.  126. 

the  scale  is  large,  it  must  be  done  ;  but  if  the  scale  is  very 
small,  the  following  is  a  sufficiently  close  approximation :  Set 
the  table  level,  with  c  over  C  and  with  cb  approximately  in  line 
with  CB.  Loosen  the  azimuth  motion  and,  with  the  alidade 
on  the  line  <$.,  bring  the  line  of  sight  on  B.  Clamp  in  azimuth, 
and  the  table  is  set.  The  point  c  is  not  exactly  over  (7,  but 
the  error  will  be  inappreciable. 

The  table  being  set  over  C  and  oriented  on  CB,  turn  the 
alidade  toward  D  and  draw  cd.  da  may  now  be  connected, 
giving  the  entire  field.  It  will  be  better  to  occupy  D  and  see 
whether  the  line  DA  will  pass  through  a  on  the  paper,  and 
whether  the  length  da  equals  to  scale  the  length  DA. 


272 


TOPOGRAPHICAL   SURVEYING. 


Another  method  very  much  used  with  the  plane  table  is 
known  as  the  method  of  intersections.  Let  it  be  required  to 
locate  the  points  A,  B,  (?,  Fig.  127,  from  the  points  D  and  E. 
Measure  DE,  and  lay  off  to  scale  on  the  paper  a  line  equal  to 
DE,  in  proper  position  so  that  the  points  desired  will  fall  on 
the  paper.  Set  the  table  over  .D  and  orient  on  E.  Swinging 
the  alidade  about  d,  draw  lines  toward  (7,  B,  and  A.  Set  the 
table  over  E  and  orient  on  D.  Swinging  the  alidade  about  e, 
draw  lines  toward  0,  B,  and  A,  and  note  the  intersections  of 
these  lines  with  those  drawn  from  d  to  corresponding  points. 
These  intersections  locate  the  points. 


256.    The  three-point  and  two-point  problems.     In  filling  in 
topography  that  is  hung  on  a  system  of  triangles,  it  is  common 
to   complete  the  triangulation  and  map  it.     There  will  usu- 
ally be  drawn  on  a  single  sheet  on  the  plane  table  only  the 
B  topography   adja- 

/  \  cent  to  one  or  two 

A^  /      \  triangulation  sta- 

i  x\  /  \  tions,     the    work 

\         \%  /  \  being    carried    on 

1  \N    /  \  precisely  as  in  the 

\  A'X  \  ^'-f         use  of  the  transit 

and  stadia,  except 
that  it  is  mapped 
at  once.  The  first 
sheet  will  ordina- 
rily have  plotted 
on  it  at  least  one 
triangulation  sta- 
tion and  a  line 
toward  another.  These,  indeed,  might  be  drawn  at  random, 
so  that  they  be  located  conveniently  on  the  sheet  for  the  work 
to  be  done.  Usually  there  will  be  two  or  more  triangulation 
stations,  or  previously  occupied  plane  table  stations,  mapped  on 
a  new  sheet.  The  table  may  be  set  over  one  and  oriented  on 
another.  It  not  unfrequently  happens  that  the  table  can  not 
conveniently  be  set  over  any  one  of  the  stations  mapped  on  the 
sheet.  Then,  if  the  table  is  set  over  any  point  at  random  and 


FIG.  127. 


THE   PLANE   TABLE. 


273 


properly  oriented  (assuming  that  the  latter  could  be  done  by 
the  needle  or  otherwise)  and  the  alidade  is  then  revolved  sepa- 
rately about  each  plotted  point  and  directed  toward  the  place  in 
the  field  thereby  represented,  a  line  being  drawn  on  the  paper 
in  this  direction,  the  lines  so  drawn  from  the  various  points 
should  all  intersect  at  one  point,  which  o 

would  be  the  properly  mapped  point  over  • 

which  the  instrument  is  set.    In  Fig.  128,  if  • 

points  A,  B,  (7,  have  been  properly  mapped  \  \ 

in  a,  5,  and  <?,  and  if  the  table  is  set  up  so     \  I 

that  ab  is  parallel  to  AS,  and  consequently        \  •  ,P 

be  is  parallel  to  BC,  and  lines  are  drawn  \        • 

through  a,  6,  and  c  in  the  directions  J.a,  \      1 

Bb,  and  Cc,  they  should  intersect  in  the 
point  d  which  is  over  D  over  which  the 
table  is  set.  The  difficulty  is  to  orient 
the  table.  This  is  done  (when  three 
points  are  mapped)  by  what  is  called  FlG-  128- 

the  "  three-point  problem  "  and,  when  two  points  are  mapped, 
by  the  "  two-point  problem,"  or  "  location  by  resection  on  the 
known  points."  There  are  several  solutions  of  these  problems. 
The  following  are  perhaps  the  simplest  and  most  satisfactory. 

(1)  Three-point  problem.     If  three  points  are  mapped  and 
may  be  seen  from  the  position  of  the  table,  place  a  piece  of 
tracing  paper  on  the  table,  and,  assuming  any  convenient  point 
on  the  tracing  paper,  draw  lines  from  this  toward  the  three 
field  points.     Remove  the  alidade  and  shift  the  tracing  paper 
till  the  three  lines  pass  through  the  three  mapped  points.     Prick 
through  the  intersection  of  the  three  lines.     This  is  the  point 
required.     Now  orient  the  table  on  any  one  of  the  points. 

(2)  Two-point  problem.     In    this  problem  two  points  are 
mapped  on  the  paper,  and  a  third  is  occupied  on  the  ground  ; 
the  position  of  this  third  point  is  to  be  correctly  mapped,  and 
the  table  oriented  so  that  work  may  proceed  from  the  third 
point.     To  do  this,  a  fourth   point   is  occupied  temporarily. 
In  Fig.  129  the  field  points  have  been  mapped  and  are  shown 
on  the  table  in  a  and  b.     The  point  O  is  to  be  occupied.     Place 
a  piece  of  tracing  paper  on  the  table  and  set  the  table  over  a 
fourth  point  D,  conveniently  chosen.     Through  the  point  on 

R'M'D  SURV.  — 18 


274 


TOPOGRAPHICAL   SURVEYING. 


FIG.  12i 


the  tracing  paper  that  is  over  D  draw  lines  toward  A,  B,  and  C. 
Estimate  the  distance  D<7,  and  lay  it  off  to  scale  to  c'.  Set  c' 
over  C  and  orient  on 
D.  Draw  lines  toward 
A  and  B,  and  note  the 
intersection  of  these 
lines  with  those  al- 
ready drawn  from  d'. 
The  figure  a'Vc'd'  will 
be  similar  to  the  fig- 
ure ABCD.  It  will, 
however,  not  be  drawn 
to  the  proper  scale 
since  the  distance  DO 
was  only  estimated. 
Therefore,  a'V  will  be 
longer  or  shorter  than  it  should  be  to  represent  properly  the 
line  AB  to  the  scale  that  it  is  proposed  to  use.  Let  it  be 
supposed  to  be  shorter.  Remove  the  alidade  and  shift  the 
tracing  paper  till  a'  is  over  a,  and  the  line  a'b' 
is  coincident  with  ab.  Construct  a  quad- 
rilateral on  ab  similar  to  a'b'c'd'.  This  will 
give  the  quadrilateral  abed,  and  c  is  properly 
located  relatively  to  a  and  b.  The  point  is 
pricked  through  into  the  drawing,  and  the 
table  is  set  up  with  c  over  C  and  oriented  on 
any  one  of  the  other  points,  and  the  work  proceeds.  The  trac- 
ing paper  may  be  dispensed  with,  the  drawing  work  being 
thereby  somewhat  increased.  This  method  is  seen  to  be 
merely  a  modification  of  the  three-point  problem. 

257.  Adjustments.  The  bubbles  for  leveling  the  table  are 
adjusted  to  be  parallel  to  the  base  of  the  alidade.  This  is  done 
by  the  method  already  given  for  adjusting  a  level  having  a 
plane  metallic  base.  See  Art.  51. 

The  line  of  collimation  and  the  axis  of  the  telescope  bubble 
are  made  parallel  by  the  "peg"  adjustments  as  applied  to  the 
transit. 

The  horizontal  axis  is  adjusted  like  that  of  the  transit. 


FIG.  130. 


CHAPTER   X. 


EARTHWORK  COMPUTATIONS. 


ORDINARY   METHODS. 


258.  Occurrence  of  problem.  The  surveyor  is  frequently 
called  upon  to  measure  the  volume  of  earth  moved  or  to  be 
moved  in  connection  with  various  grading  operations.  These 
may  consist  of  the  grading  of  a  city  block  to  conform  to  the 
bounding  street  grades,  the  simple  excavation  of  a  foundation 
pit  or  cellar,  the  grading  of  a  street,  the  building  of  a  reservoir, 
the  dredging  of  material  from  the  bot- 
tom of  a  river,  lake,  or  bay,  etc. 

Payment  for  earthwork  is  usually 
by  the  cubic  yard  ;  sometimes,  for 
small  quantities,  by  the  cubic  foot. 
The  measurement  of  earthwork  is 
based  on  the  geometrical  solids,  the 
prism,  wedge,  pyramid,  and  prismoid. 


259.  Prisms.  The  volume  of  any 
doubly  truncated  prism  is  V=A A,  in 
which  A  is  the  area  of  a  right  section, 
and  A  is  the  element  of  length  through 
the  center  of  gravity  of  the  right  sec- 
tion or  is  the  length  of  the  line  joining 
the  centers  of  gravity  of  the  two  ends.  FJG  131 

In  any  prism  having  a  symmetrical 

right  base,  the  center  of  gravity  of  that  base  is  at  its  center  of 
form.  This  is  true  of  all  the  regular  polygons  and  of  all 
parallelograms. 

In  such  prisms  the  length  of  the  element  through  the  center 
276 


276 


EARTHWORK   COMPUTATIONS. 


of  gravity  is  the  mean  of  all  the  edges.     This  is  further  true  of 
a  triangular  prism. 

This  may  be  shown  as  follows  : 

In  the  truncated  triangular  prism  shown  in  Fig.  131,  the 

2A 
area  of  the  right  base  is  A  =  ^  bp,  and  p  =  —  .     The  volume  is 

-         -  Substituting  for  f  ils 


value   **     7= 
o 


3 


A  truncated  parallelepiped  may  be  divided  into  two  trun- 
cated triangular  prisms  in  two  ways,  as 
shown  in  Fig.  132,  and  each  set  may  be 
treated  as  above  and  the  resulting  volumes 
added.  This  gives  twice  the  volume  of 
the  solid,  which  divided  by  two  gives 

A   -f  A 


, 


FIG.  132. 


A  prism  having  for  its  right  base  a 
regular  pentagon  may  be  divided  into 
three  triangular  truncated  prisms  in  five 
ways  and  treated  as  the  parallelepiped, 

when  the  result  r=A  +  *«  +  *s  +  *4  +  ^ 

5 
is  obtained. 


260.  Prismoids.  Many  of  the  forms  dealt  with  in  earth- 
work are  more  nearly  prismoids  than  either  prisms,  wedges,  or 
pyramids,  and  when  this  is  the  case  the  prisrnoidal  formula 
gives  more  correct  volumes. 

A  prismoid  is  a  solid  having  two  parallel  polygonal  bases 
connected  by  triangular  faces.  This  is  equivalent,  if  the  faces 
are  conceived  small  enough,  to  the  following,  which  somewhat 
better  defines  the  forms  dealt  with  in  earthwork  : 

A  prismoid  is  a  solid  having  parallel  plane  ends  with  sides 
formed  by  moving  a  ruled  line  around  the  perimeters  of  the 
ends  as  directrices.  Such  a  solid  may  be  conceived  as  made  up 
of  a  combination  of  prisms,  pyramids,  and  wedges. 


i  The  student  should  show  that 
the  center  of  gravity  of  the  bases. 


js  the  length  of  the  element  through 


ORDINARY  METHODS. 


277 


261.  Prismoidal  formula.  If  a  single  expression  can  be 
found  giving  the  volume  of  a  prism,  pyramid,  and  wedge,  it 
will  give  the  volume  of  a  prismoid,  since  such  a  solid  is  made 
up  of  the  above  three  elementary  solids,  all  having,  in  any 
given  case,  equal  heights. 

In  the  right 


prism 


pyramid 


F=      ~ 


Ah 

1 

Ah 
3 

which  may  be 
written 

h 
6 

Ah 

h 

2 

~Q 

in  which  A-^  is  one  base,  A2  is  the  other  base,  and  Am  is  a  section 
midway  between  the  two  end  bases  and  parallel  to  them. 

The  single  expression  thus  found  is  called  the  prismoidal 
formula,  and  is  the  correct  formula  for  finding  the  volumes  of 
prismoids. 

Am  is  not  a  mean  of  A^  and  Av  but  each  of  its  linear  dimen- 
sions is  a  mean  of  the  corresponding  dimensions  of  A1  and  A2. 

This  formula  is  applicable  to  the  sphere  and  all  the  regular 
solids  of  revolution.1 

262.  Approximations.  To  avoid  the  labor  involved  in  com- 
puting the  volume  of  prismoids  by  the  prismoidal  formula,  one 
of  two  assumptions  is  frequently  made.  These  are  : 

(1)  The  volume   of   a  prismoid   is      *  "|"     2A,  called   the 

method  of  average  end  areas. 

(2)  The  volume  of  a  prismoid  is  hAm,  called  the  method 
of  mean  areas. 

The  error  of  the  first  assumption  is  twice  that  of  the  second 
and  of  the  opposite  sign.  To  show  this,  let  av  av  and  a.,  be 
homologous  sides  of  the  similar  end  and  middle  sections  of  a 
given  prismoid.2  Then  since  the  areas  of  similar  figures  are  as 

1  Let  the  student  show  the  truth  of  this  statement  for  the  sphere  and  both 
spheroids  of  revolution. 

2  Prismoid  end  areas  are  not  necessarily  similar.    They  are  here  assumed  to  be 
so  for  simplicity. 


278  EARTHWORK  COMPUTATIONS. 

the  squares  of  their  homologous  sides,  there  may  be  written  the 
following  expressions  for  the  volume  of  the  prismoid,  based  on 
the  true  and  approximate  formulas.  In  these  expressions  K 
is  any  constant  ratio  by  which  a2  is  to  be  multiplied  to  give 

area.     Remembering  that  am  =  a*  j"  a2, 
By  prismoidal  formula 

Vp  =  2  K±  (a*  +  a,a,  +  af)  =  ±Kh  (a*  +  a,a2  +  a*).      (1) 
By  average  end  areas 


V"!       i      -a  y 

/• 

By  mean  areas 
Subtracting  (1)  from  (2), 

Vea  —    Vp  =       ^      ^  ""  a2-^2'  (^ 

Subtracting  (!')  from  (3), 

It  is  seen  that  equation  (5)  is  half  of  equation  (4)  and  of 
opposite  sign.  From  equations  (4)  and  (5)  it  is  also  seen 
that  the  average  end  area  method  gives  results  too  great,  while 
the  mean  area  method  gives  results  too  small.  There  are  cer- 
tain peculiar  cases  when  the  end  areas  are  not  similar  figures  in 
which  the  average  end  area  method  gives  results  too  small. 

By  the  prismoidal  formula,  or  one  of  the  approximations 
mentioned,  the  volume  of  all  masonry  work  or  earth  work  may 
be  computed. 

It  is  seen  from  equations  (4)  and  (5)  that  the  errors  of  the 
assumptions  given  vary  with  the  square  of  the  difference  in 
dimensions  of  the  two  end  areas,  and  therefore  it  may  be  con- 
cluded that  in  cheap  work,  such  as  earthwork,  if  successive 
end  areas  are  nearly  alike,  the  simpler  method  of  average  end 


ORDINARY  METHODS. 


279 


areas  may  be  used,  while  if  the  areas  are  quite  different,  the 
prismoidal  formula  should  be  used  ;  and  in  costly  work,  such 
as  masonry,  the  prismoidal  formula  should  always  be  used. 

263.  Area  grading.  When  a  large  area,  as  a  city  block,  is 
to  be  filled  or  excavated,  the  area  may  be  divided  into  rectan- 
gles of  such  size  that  the  four  cor- 
ners of  each  may  be  assumed  to  be 
in  one  plane.  The  lines  of  division 
will  be  so  referenced  that  their  in- 
tersections can  be  again  found  after 
the  grading  is  done.  Elevations  of 
the  corners  of  the  rectangles  will 
be  found  before  grading  and  after 
the  grading  is  done.  The  difference 
in  elevation  at  any  one  corner  is  the 
depth  of  cut  or  fill  at  that  corner.  FIG.  133. 

The  depth  of  cut  or  fill  is  marked  on  the  stakes  set  at  the  cor- 
ners to  guide  the  workmen.  The  total  volume  is  thus  divided 
into  a  number  of  prisms  of  equal  bases  and  known  altitudes. 
The  volume  in  cubic  yards  of  one  prism  is,  letting  A  be  its  base 
and  a,  6,  c,  c?,  its  four  corner  heights, 


2        1 

c      h 

i 

i 

i 

1 

f      i 

Z 

* 

4 

i 

I 

* 

i       2 

2 

i 

i 

i 

1 

1        2 

3 

] 

1        2 

I 
Z 

i 

2 

2 

i 

1 

D 

27 


The  student  may  show  that  the  volume  of  all  the  prisms  is 
given  by 

V '= —  (2Aj  -f-  2  2A2  +  3  2A3  +  4  2^4), 

4  x  27 

in  which  2  is  the  sign  for  "  sum  of  "  and  hv  A2,  etc.,  are  the  cor- 
ner heights  of  the  several  prisms,  the  subscript,  numeral  indi- 
cating the  number  of  prisms  of  which  the  h  to  which  it  is 
affixed  is  one  corner.  Thus  c  would  be  an  A4,  b  an  A2,  etc. 

The  rectangles  may  be  made  larger  by  conceiving  a  diag- 
onal as  drawn  in  each  rectangle.  This  would  be  drawn  in  the 
correct  direction  by  inspection  in  the  field,  so  that  the  assumed 
plane  tops  of  the  triangular  prisms  thus  formed  would  most 
nearly  correspond  with  the  ground  surface. 

Assuming  that  A  is  still  the  area  of  a  rectangle,  the  student 
should  show  that  the  volume  of  the  series  of  prisms  is  given  by. 


280 


EARTHWORK   COMPUTATIONS. 


O  X  ^  i 


+ 


,  etc.), 


and  that  the  highest' subscript  may  be  8. 

The  sides  of  such  rectangles  as  have  been  mentioned  may 
vary  from  twenty  feet  to  one  hundred  feet,  according  to  the 
character  of  the  ground  surface. 

264.  Street  grading.  In  grading  streets  the  sides  are 
usually  cut  down  or  filled  up  vertical  where  possible,  the  prin- 
ciple being  that  the  adjacent  property  must  look  out  for  itself. 
The  best  way  to  get  the  volume  in  street  grading  is  to  make 
cross  profiles  of  the  section  to  be  graded,  at  such  intervals  as 
may  be  necessitated  by  the  character  of  the  ground,  say  from 
fifty  to  one  hundred  feet  apart.  On  these  cross  profiles  will  be 
drawn  the  cross  profile  of  the  finished  street,  and  the  area 
between  the  two  profiles  will  be  the  area  in  cut  or  fill  at  that 
section.  The  volume  between  any  two  adjacent  sections  is 
computed  as  a  prismoid.  The  areas  may  be  drawn  to  scale 
on  cross-section  paper,  and  measured  with  a  planimeter,  or 
they  may  be  computed. 


FIG.  134. 


Some  engineers  work  with  the  grade  of  the  center  of  the 
street,  some  with  the  grade  of  the  curb  lines,  and  some  with  the 
grade  of  the  property  lines.  Stakes  are  usually  set  at  the  cen- 
ter and  sides  on  which  are  marked  the  depths  of  cut  or  fill  as 
a  guide  to  the  workmen.  Where  this  is  not  possible,  a  list  of 
cuttings  may  be  furnished  the  contractor.  Assuming  the  grade 
of  the  property  line  to  be  used  in  Fig.  134,  the  depths  of  cut 
to  the  line  AB  will  be  determined  at  such  points  in  the  cross 
section  as  may  be  necessary  to  give  a  correct  area.  The  area 
above  the  line  AB  is  computed  from  the  field  notes  and  used 


ORDINARY   METHODS.  281 

in  getting  the  volume.  The  volume  of  the  cutting  below  the 
line  AB  is  a  constant  per  unit  of  length  of  street,  and  may  be 
readily  computed  for  any  given  stretch.  The  slide  rule  is  very 
useful  in  saving  time  in  all  volumetric  computations. 

265.  Excavation  under  water.  When  excavation  under 
water  is  to  be  measured,  it  may  be  done  by  sounding  on 
known  lines,  both  before  and  after  the  excavating  is  done, 
and  the  volume  is  computed  as  in  surface  grading.  This  is 
not  always  the  best  method.  The  material  is  usually  meas- 
ured after  removal.  If  it  is  placed  in  scows,  the  displacement 
of  the  scow  for  each  load  is  obtained,  and  this,  divided  by 
the  specific  gravity  of  the  material,  gives  the  volume  in  the 
scow.  It  is  necessary  to  observe  the  displacement  of  the  scow 
for  no  load,  and  subtract  this  from  the  observed  displacement 
for  a  given  load. 

The  displacement  is  obtained  by  measuring  the  model  lines 
of  the  vessel  at  varying  depths,  and  computing  the  volume  for 
these  depths  by  the  prismoidal  formula.  These  volumes  may 
be  plotted  on  cross-section  paper  by  laying  off  the  depths  to 
scale  on  one  side  of  the  paper,  and  at  these  points  laying  off 
the  computed  volumes  to  scale  perpendicular  to  this  axis  of 
depths.  A  curve  may  now  be  drawn  through  the  points  thus 
plotted.  Ordinates  to  this  curve,  at  any  scaled  depth,  will  be 
to  scale  the  volume  for  that  depth. 

To  observe  the  displacement  correctly,  the  water  in  the  hold 
must  be  at  the  level  obtaining  when  the  displacement  of  the 
empty  vessel  was  noted.  The  depth  of  the  vessel  in  the  water 
is  best  noted  at  four  symmetrically  situated  points,  and  the 
mean  is  taken. 


ESTIMATING   VOLUMES   FROM   A   MAP. 

266.  A  reservoir.  Fig.  135  is  a  contour  map  of  a  portion 
of  a  valley  or  ravine  that  it  is  proposed  to  convert  into  a  stor- 
age reservoir  by  the  construction  of  a  dam  across  the  narrow 
part  shown.  It  is  required  to  determine  the  capacity  of  the 
reservoir.  At  the  location  of  the  dam,  its  top  width  and  side 
slopes  being  determined,  the  lines  representing  the  top,  and 


282 


EARTHWORK   COMPUTATIONS. 


what  will  be  contour  lines  of  the  finished  surface,  may  be  drawn 
on  the  map  as  indicated.     The  intersection  of  any  contour  with 
the  corresponding  contour  of  the 
finished  dam  will  be  a  point  where 
the  dam  will   join  the   side  of 
the   valley,    and   a   line,    abc, 
connecting  such  points  of  in- 
tersection will  indicate  the 
edge  of  the  proposed  struc- 
ture, or  will  be  a  line 
of  no  cut  or  fill,  that 
is,    a    "grade    line." 
The    storage    space 
may  be  conceived 
to     be     divided 
into     a     series 
of     horizontal 
layers      whose 
top    and    bot- 
tom areas   are 
the  closed  fig- 
ures      formed 
by  correspond- 
ing     contours 
of  surface  and 
dam.        These 
areas  may  be  meas- 
ured  with    a    plani- 
meter.        The     vol- 
ume of  any  layer  is 
obtained  by  averag- 
ing its  two  end  areas 
and    multiplying   by 
its  height,  which  is 
the  contour  interval. 
If  h  is  that  interval, 
the  volume  of  a  series 
of  layers   would    be 
given  by 


ESTIMATING   VOLUMES   FROM   A   MAP.  283 


If  it  is  preferred  to  use  the  prismoidal  formula,  the  height  of 
each  prismoid  will  be  twice  the  contour  interval,  and  every 
other  area  will  be  a  middle  area,  and  the  volume  of  the  whole, 
assuming  an  even  number  of  layers,  will  be 


If  there  is  an  odd  number  of  layers,  the  final  layer  may  be 
computed  separately,  either  by  average  end  areas  or  by  inter- 
polating a-  middle  area  on  the  map  and  measuring  it  with  the 
planimeter. 

The  volume  of  material  in  the  dam  may  be  obtained  in  the 
same  way.1 

267.  Application  to  surface  grading.  This  method  is  appli- 
cable also  to  the  measurement  of  irregular  earthwork  in  general. 

Let  Fig.  136  represent  a  rectangular  area  that  it  is  pro- 
posed to  surface  as  indicated.  The  full  irregular  lines  repre- 
sent contour  lines  of  the  original  surface.  The  more  regular 
curved  full  lines  represent  contours  of  the  surface  as  it  is 
proposed  to  have  it.  The  figures  represent  elevations  above 
some  datum  surface.  Conceive  the  original  and  graded  surface 
to  exist  at  the  same  time,  and  imagine  the  contour  planes 
passed  into  the  hill  at  their  respective  elevations,  and  consider 
particularly  the  82-foot  plane.  The  areas  a,  c,  b  and  w,  o,  p 
will  be  in  excavation,  while  the  areas  6,  s,  n  and  jt>,  q,  r  will  be 
in  embankment.  The  points  tf,  J,  n,  p,  and  r  are  "at  grade," 
being  neither  in  cut  nor  fill. 

Connecting  corresponding  grade  points,  there  result  the 
dotted  grade  lines  84,  /,  b,  81 ;  £,  w,  w ;  w,  p,  v,  which  are 
bounding  lines  of  bodies  of  cut  and  fill. 

Thus,  the  whole  central  portion  of  the  figure  is  in  excava- 
tion, while  the  upper  left  corner  is  in  embankment. 

The  horizontal  area  in  excavation  or  embankment  at  any  level 
is  the  area  included  between  original  and  finished  surface  contours 
at  that  level. 

1  The  student  should  show  how  to  do  this,  using  Fig.  135. 


284 


EARTHWORK  COMPUTATIONS. 


To  get  the  volume  of  any  single  body  of  cut  or  fill,  measure 
successive  areas  with  the  planimeter,  and  assume  these  to  be 
end  areas  of  figures  that  are  as  nearly  prismoids  as  anything 
else,  with  altitudes  equal  to  the  contour  interval.  If  the  pris- 
moidal  formula  is  to  be  used  for  computing  the  volume,  the 

80  81  82          83         84  S5U    t    86  87 


FIG.  136. 

altitude  of  a  single  prismoid  is  taken  as  twice  the  contour 
interval,  that  is,  two  layers  are  taken  to  make  one  prismoid. 
The  volumes  may  be  computed  by  the  formulas  of  the  last 
article.  Assuming  average  end  areas,  the  small  body  of  cut  on 
the  right  is  found  to  have  areas  : 

At  elevation  81,  00. 

At  elevation  82,  acb. 

At  elevation  83,  def. 

At  elevation  84.  00. 


Volume  = 


+  def  +         = 


ESTIMATING  VOLUMES   FROM   A  MAP. 


285 


The  portion  of  fill  between  this  cut  and  the  large  central 
body  of  cut  has  areas: 

At  88-foot  level,  00. 

At  87-foot  level,  ghi. 

At  86-foot  level,  klm. 

etc.,      etc. 

268.  Application  to  structures.  One  example  of  the  appli- 
cation of  this  method  of  estimating  volumes  to  regular  struc- 
tures built  on  irregular  ground  will  be  given.  The  form  of  the 


FIG.  137. 

structure  is  not  presented  as  an  example  of  good  engineering, 
but  merely  to  show  the  method  of  estimating  volumes. 

A  small  reservoir  is  to  be  built  on  a  hillside,  and  will  be 
partly  in  excavation  and  partly  in  embankment.  Fig.  137 
shows  such  a  case.  The  contours,  for  the  sake  of  simplicity, 
are  spaced  five  feet  apart.  The  top  of  the  reservoir  (shown  by 


286  EARTHWORK   COMPUTATIONS. 

the  heavy  lines  making  the  square)  is  10  feet  wide,  and  at 
an  elevation  of  660  feet.  The  reservoir  is  20  feet  deep,  with  side 
slopes  —  both  inside  and  outside  —  of  two  to  one,  making  the 
bottom  elevation  640  feet,  and  20  feet  square,  the  top  being  100 
feet  square  on  the  inside.  The  dotted  lines  are  contours  that 
would  be  invisible  if  both  original  surface  and  completed  reser- 
voir were  supposed  to  exist  at  the  same  time.  The  areas  of  fill 
all  fall  within  the  broken  line  marked  abcdefghik,  and  the  cut 
areas  all  fall  within  the  broken  line  marked  abcdefyo.  These 
broken  lines  are  grade  lines.  The  areas  of  fill  and  cut  are 
readily  traced  by  following  the  closed  figures  formed  by  con- 
tours of  equal  elevation: 

At  640-foot  level  area  in  fill  is  pst. 
At  650-foot  level  area  in  fill  is  Imnuvxl. 
At  650-foot  level  area  in  cut  is  1  2  3  ux  1, 

The  other  areas  are  easily  traced.  In  the  figures  given,  the 
lines  have  all  been  drawn  in  black  for  printing.  In  practice 
they  should  be  drawn  in  different  colors  to  avoid  confusion. 


CHAPTER  XL 

HYDROGRAPHIC  SURVEYING. 

269.  Definition.    A  hydrographic  survey  is  a  survey  having 
to  do  with  any  body  of  water.     A  topographic  survey  may  be, 
and  frequently  is,  partly  a  hydrographic  survey. 

270.  Objects.     A  hydrographic  survey  may  be  undertaken 
for  any  one  of  the  following  purposes  : 

(1)  To  determine  the  topography  of  a  portion  of  the  bed  of 
the  sea,  a  bay,  or  harbor,  or  river,  in  order  that   it   may  be 
mapped   for   the   information   of   seamen.       In  this  case  it  is 
necessary  merely  to  locate  the  channels,  dangerous  rocks,  and 
shoals. 

(2)  To  determine  definitely  the  configuration  of  a  small 
portion  of  the  bed  of  the  sea,  a  bay,  harbor,  or  river,  for  the 
purpose  of  planning  works  to  rest  on  or  in  the  bed,  such  as 
lighthouses,  sea  walls  or  docks,  bridge  piers,  etc.,  or  to  esti- 
mate the  quantities  of  materials  to  be  moved  by  dredging  or 
blasting,  for  the  improvement  of  the  channels  or  harbor. 

(3)  To  determine  the  flow  of  a  given  river  or  estuary  and 
the  direction  of  the  currents,  for  the  purpose  of  studying  the 
physics  of  the  stream  or  for  planning  waterworks  or  drainage 
systems. 

(4)  To  secure  the  necessary  information  for  planning  the 
improvement  of  marshy  shores  of  a  lake  or  stream  by  lowering 
the  water  level  or  otherwise. 

271.  Work   of   the   surveyor.     The   survey  ends  with   the 
determination  of  the  required  information.     The  planning  of 
the  works  and  their  execution  is  hydraulic  engineering.     The 
laying  out  of  the  plan  on  the  ground  requires  the  methods  of 
the  surveyor.     The  work  of  the  surveyor,  then,  is  to  make  a 

287 


288  HYDROGRAPHIC   SURVEYING. 

topographical  map  of  the  area  to  be  covered,  to  compute  mate- 
rial moved,  to  determine  cross  sections  of  streams,  their  ve- 
locities, their  discharge,  the  direction  of  their  currents,  and  the 
character  of  their  beds,  and  to  lay  out  projected  improvements. 

272.  General  statement  of  methods.  The  configuration  or 
topography  of  the  bed  of  a  body  of  water  is  determined  by 
sounding,  that  is,  measuring  the  depth  of  water.  If  many  points 
are  observed,  a  contour  map  of  the  bottom  may  be  drawn,  the 
water  surface  being  the  plane  of  reference. 

The  cross  section  of  a  stream  is  determined  by  taking 
soundings  along  a  fixed  transverse  line. 

The  velocity  of  flow  of  a  stream  is  determined  by  noting 
how  long  a  float  requires  to  drift  a  known  distance,  or  by 
the  use  of  a  current  meter. 

Since  the  velocity  is  not  the  same  at  all  parts  of  a  given 
cross  section,  many  determinations  for  velocity  must  be  made 
before  an  average  for  the  whole  cross  section  can  be  obtained. 

The  discharge  of  a  large  stream  like  the  Mississippi,  Mis- 
souri, Ohio,  or  Hudson  River  is  determined  by  measuring  a 
cross  section  and  finding  the  velocity  of  flow  past  it.  The 
discharge  of  a  small  stream,  like  any  one  of  the  thousands 
of  creeks,  is  obtained  by  weir  measurements.  A  weir  is  a 
notch  cut  in  a  dam.  A  dam  is  constructed  across  the  stream, 
and  a  notch  is  built  in  it,  through  which  the  water  flows. 
From  the  known  size  of  the  notch  and  the  depth  of  water 
flowing  over  it,  the  flow  may  be  calculated  by  methods  that 
are  explained  in  full  in  any  work  on  hydraulics.  Insignificant 
streams  may  be  measured  by  any  simple  means  that  may 
occur  to  the  surveyor. 

The  direction  of  surface  currents  may  be  determined  by 
means  of  floats.  Subcurrents  are  best  determined  by  means 
of  the  direction  meter. 

The  character  of  the  bed  of  a  body  of  water  may  be  ascer- 
tained by  securing  samples  in  a  cup  attached  to  the  sounding 
lead,  or  by  putting  tallow  in  a  cavity  in  the  bottom  of  the  lead. 
Some  of  the  bottom  will  adhere  to  the  tallow.  If  the  character 
of  the  bed  for  some  distance  below  the  bottom  is  required,  as 
when  bridge  piers  are  to  be  founded  on  firm  bottom  which  is 


SOUNDINGS. 


289 


at  an  unknown  depth,  various  methods  are  resorted  to,  accord- 
ing to  the  depth  of  water,  the  soil  below,  and  the  difficulties 
to  be  overcome.  Simple  cases  may  be  handled  with  gaspipe 
rods  driven  through  the  bottom  by  hammers  to  a  hard  layer 
of  material.  In  other  cases  drills  driven  by  pile  drivers  are 
used,  or  diamond  drills  if  rock  is  encountered,  and  it  is  neces- 
sary to  determine  its  character  and  depth. 

The  method  of  making  weir  measurements,  being  properly 
a  part  of  the  study  of  hydraulics,  and  requiring  considerable 
space  to  treat  adequately,  will  not  be  given  here ;  but  such 
methods  as  require  the  use  of  surveying  instruments  and  meters 
will  be  described  in  sufficient  detail  to  enable  the  student  to 
undertake  such  work. 

SOUNDINGS. 

273.  Making  soundings.  Deep-sea  soundings  are  made  with 
special  elaborate  apparatus  which  will  not  be  described. 

Soundings  in  moderately  deep  water  are  made  with  a  weight, 
known  as  a  lead,  attached  to  a  suitable  line. 

For  depths  of  less  than  fifteen  or  twenty  feet  a  pole  is  best. 

The  lead  may  be  any  heavy  weight.  It  is  preferably  of 
lead,  molded  about  an  iron  rod.  The  form  is  shown  in  Fig. 
138.  The  cup  at  the  bottom  is  for  collecting  samples 
of  the  bottom.  There  is  a  leather  washer  that  slides 
up  and  down  the  rod  between  the  bottom  of  the  lead 
and  the  top  of  the  cup.  This  keeps  the  soil  that  is 
collected,  in  the  cup,  while  the  lead  is  being  raised. 
Very  often  the  cup  is  omitted,  and  if  samples  of 
the  bottom  are  required,  tallow  is  used  as  already 
described. 

The  line  for  very  deep  work  may  be  of  wire,  but 
must  be  used  with  a  reel.  For  ordinary  work,  a 
hemp  line  or  a  chain  is  best.  A  line  must  be 
stretched,  but  not  too  much,  as  it  may  shrink.  It 
must  be  frequently  tested. 

For  soundings  for  navigation  charts,  the  depths  are  taken 

in  feet  to  four  fathoms,  and  thereafter  in  fathoms.     The  party 

required  to  make  soundings  consists  of  a  leadsman,  a  recorder, 

one  or  more  oarsmen,  and,  when  soundings  are  located  from  a 

R'M'D  SURV.  — 19 


FIG. 


290  HYDROGRAPHIC   SURVEYING. 

boat,  one  or  two  observers  to  read  the  angles.  If  the  boat  is 
located  by  angles  read  on  shore,  there  must  be  one  or  more 
observers  on  shore. 

The  leadsman  casts  the  lead  and  announces  the  depths.  For 
this  purpose  the  line  is  graduated  by  suitable  tags,  so  that  the 
depths  can  be  readily  determined.  In  still,  deep  water  the  boat 
may  be  stopped  while  the  sounding  is  being  takes  and  recorded, 
but  more  often  it  is  not  stopped.  The  leadsman  stands  in  the 
bow  of  the  boat  and  throws  his  lead  ahead,  so  that,  as  he  judges, 
it  will  be  on  the  bottom  and  the  line  will  be  vertical  when  the 
boat  is  over  it.  He  will  become  expert  at  this  after  a  time. 
The  recorder  notes  the  number  of  the  sounding  and  the  depth 
that  the  leadsman  gives  him. 

The  method  of  using  rods  will  be  obvious. 

The  reference  plane  for  depths  is  the  surface  of  the  water, 
and  as  this  is  continually  changing,  except  in  still  ponds  or 
lakes,  it  is  necessary  to  know  the  stage  of  the  water  at  the  time 
the  soundings  are  taken.  This  is  done  by  establishing  a  gage 
in  the  vicinity  of  the  work.  The  soundings  on  all  navigation 
charts  are  referred  to  mean  low  tide.  The  zero  of  the  gage 
should  be  set,  if  possible,  at  this  elevation.  The  gage  may  be 
a  graduated  board  nailed  to  a  pile  in  a  pier  in  the  vicinity  of 
the  work  and  read  at  hourly  intervals  during  the  progress  of 
the  sounding.  Gages  for  rivers  are  frequently  inclined,  being 
laid  along  the  shore  at  right  angles  to  the  stream,  and  the 
points  of  equal  differences  of  altitude  are  determined  by  level- 
ing. A  tide  gage  that  is  to  remain  permanently  in  place  should 
be  self -registering.  Such  a  gage  may  consist  essentially  of  a 
float  protected  by  a  surrounding  house  or  tube,  and  attached 
by  suitable  mechanism  to  a  pencil  that  has  a  motion  propor- 
tional to  the  rise  and  fall  of  the  float.  The  pencil  bears  against 
a  piece  of  graduated  paper  fastened  to  a  drum  that  is  revolved 
by  clockwork.  There  will  thus  be  drawn  on  the  paper  a  pro- 
file in  which  the  horizontal,  units  are  time,  and  the  vertical 
units  are  feet,  rise,  and  fall.  The  stage  of  the  tide  for  any 
instant  can  be  read  from  the  profile. 

274.  Locating  the  soundings.  (1)  The  soundings  may  be 
located  by  two  angles  read  simultaneously  from  the  opposite 


SOUNDINGS. 


291 


ends  of  a  line  on  shore,  the  recorder  in  the  boat  signaling  the 
observers  on  shore  when  a  sounding  is  about  to  be  taken,  and 
again  when  it  is  taken. 

(2)  The  soundings  may  be  located  by  establishing  two  flags 
on  shore  in  a  line  nearly  normal  to  the  shore  line  and  taking  the 


Fto.  139. 

soundings  in  line  with  the  range  thus  formed,  the  position  of  the 
boat  being  determined  by  an  angle  read  on  shore,  or  by  time  in- 
tervals. A  series  of  such  ranges  is  laid  out,  and  soundings  are 
taken  along  each  range. 
The  point  selected  for  . 
occupation  by  the  angle-  ^s> 
measuring  instrument  on 
shore  should  be  so  chosen 

that  the  angles  of  inter-  » 

section  with  the  range 
lines  will  be  large  enough 
to  give  good  locations. 
If  the  shore  is  narrow 
and  precipitous,  the 
range  may  be  established 
by  placing  a  flag  on 
shore  and  a  buoy  in  the 
water.  The  position  of  the  buoy  is  determined  as  was  described 
in  the  first  method  for  locating  soundings.  The  distances  be- 
tween the  shore  points  will  be  measured  for  base  lines,  and  the 
angles  will  be  measured  from  these  base  lines.  Fig.  139  will 
explain  the  method.  The  buoys  may  be  any  visible  float,  an- 
chored in  place  by  any  heavy  weight  attached  to  the  buoy  by  a 
rope.  The  rope  must  be  sufficiently  long  to  permit  the  buoy  to 


FIG.  140. 


292 


HYDROGRAPHIC   SURVEYING. 


be  seen  at  high  tide.  At  low  tide  the  position  of  the  buoy  will 
not  be  certain,  for  obvious  reasons.  A  good  kind  of  buoy  is  a 
tapering  spar  of  wood.  If  it  is  not  readily  visible,  a  flag  may 
be  set  in  the  end. 

(3)  If  many  soundings  are  to  be  taken  at  different  times  on 
one  cross  section  of  a  river,  they  may  be  located  by  taking 
them  at  the  intersection  of  a  series  of  ranges  so  laid  out  as  to 
have  their  intersections  on  the  required  cross  section.     Fig.  140 
will  sufficiently  explain. 

(4)  Again,  the  soundings  may  be  located  by  stretching  a 
rope  or  wire  across  the  channel  and  taking  them  at  marked 


FIG.  141. 

intervals  along  the  line.  This  method  is  adapted  to  narrow 
channels,  and  is  used  somewhat  in  connection  with  measure- 
ment of  materials  dredged  in  such  channels. 

(5)  Again,  they  may  be  located  by  measuring  in  the  boat, 
at  the  time  the  soundings  are  taken,  two  angles  to  three  known 
points  on  shore.  The  angles  are  measured  with  a  sextant. 
There  are  several  methods  of  obtaining  from  the  data  thus 
secured  the  location  of  the  point  where  the  sounding  is  taken. 
The  problem  to  be  solved  is  known  as  the  "three-point  prob- 
lem." 

The  most  convenient  way  to  plot  the  point  on  a  map  on 
which  have  been  already  plotted  the  known  shore  points,  is  by 
the  use  of  a  three-arm  protractor,  shown  in  Fig.  141.  The  two 


SOUNDINGS. 


measured  angles  are  set  on  the  protractor  and  the  arms  aro 
made  to  coincide  with  the  plotted  points.  The  center  then 
gives  the  required  position  of  the  sounding. 

If  no  three-arm  protractor  is  available,  draw  three  lines  on 
a  piece  of  tracing  paper  so  that  the  two  measured  angles  are 
included  between  the  lines.      Shift  the  paper  over  the  map 
till  the  three  lines  pass 
through  the  three  points. 
The   point    from   which 
they  radiate  is  now  over 
the  position  of  the  sound- 
ing, and  may  be  pricked 
through. 

The  position  of  the 
point  may  be  found  al- 
gebraically as  follows  : 
In  Fig.  142,  the  known 
points  are  A,  B,  C,  de- 
termined by  the  angle  A, 
and  the  sides  a  and  b. 
The  measured  angles  are 
a  and  /3,  and  the  required  point  is  P. 
and  GPB  there  is  obtained 


FIG.  142. 


From  the  triangles  APB 


PB  = 


a  sin  I      b  sin  m 


Also, 

whence, 

and 


sin  a          sin  /? 
I  +  m  =  360°  -  (A  +  « 

m  =  S  -  I, 
sin  m  —  sin  S  cos  I  —  cos  S  sin  I. 


(1) 
(2) 

(3) 


From  (1)  and  (3)  determine  Z,  which  being  found,  the  triangle 
APB  may  be  solved  and  P  located.  To  solve  for  I,  substitute 
in  (1)  the  value  of  sin  m  found  in  (3),  reduce  to  common 
denominator,  divide  by  sin  Z,  transpose,  and  get 


cot  I  = 


I  sin  «  sin  S 


=  cot  s 


b  sin  «  cos  S 


•(4) 


275.    Occurrence   of   methods.      The   fifth   method    is    used 
almost  exclusively  in  locating  soundings  in  bays,  harbors,  and 


294  HYDROGRAPHIC    SURVEYING. 

off  the  seacoast.     The  others  are  used  in  connection  with  river 
and  small  lake  surveys. 

276.  Survey  of  a  harbor,  river,  etc.     In  making  a  complete 
map  of  a  harbor,  coast,  or  river,  a  system  of  triangles  of  greater 
or   less   extent   is   laid   out.     The  triangulation  stations  thus 
locate  the  prominent  points.     These  points  are  then  used  as  a 
basis  for  locating  the  soundings.     If  too  far  apart,  other  points 
are  located  along  shore  by  running  a  traverse  between  triangu- 
lation stations.     Such  a  traverse  is  usually  run  in  any  event  to 
determine  the  detail   configuration  of   the  shore  line  and  the 
topography  adjacent  thereto.     This  work  is  best  done  with  the 
transit  and  stadia  or  with  the  plane  table. 

The  map  is  finished  like  any  topographical  map.  The 
contour  lines  of  the  bottom  are  frequently  drawn  as  dotted 
lines  to  a  depth  of  four  fathoms,  and  beyond  that  depth  the 
soundings  are  given  in  figures  expressing  fathoms  and  quarters. 
Charts  of  the  various  harbors  and  many  portions  of  the  coast 
of  this  country  are  published  by  the  United  States  Coast  and 
Geodetic  Survey,  and  may  be  purchased  for  a  small  price.  On 
these  charts  are  located  buoys  and  beacons.  These  are  estab- 
lished to  denote  shoals  and  rocks,  and  are  so  arranged  that  in 
entering  a  harbor  a  red  buoy  with  even  number  is  to  be  passed 
on  the  right;  a  black  buoy  with  odd  number  is  to  be  passed  on 
the  left,  and  buoys  with  red  and  black  stripes  may  be  passed 
on  either  the  right  or  the  left.  Buoys  in  channels  are  painted 
with  black  and  white  vertical  stripes. 

Beacons  and  buoys  are  different  things.  A  beacon  is  a  per- 
manent fixed  signal,  usually  on  a  shoal  or  dangerous  rock  ; 
while  a  buoy  is  a  float  of  some  kind,  anchored  by  a  chain.  It 
is  used  to  denote  either  danger  or  channel. 

THE  SEXTANT. 

277.  Description.     It  has   been  said  that  when  the  angles 
are  read  in  the  boat,  they  are  measured  with  a  sextant.     The 
sextant  is  shown  in  Fig.  143.     It  consists  of  an  arc  of  sixty 
degrees,  but,  from   the   principle    of   the  instrument,  reading 
angles  up  to  one  hundred  and  twenty  degrees.     I  is  a  mirror 
attached  to  the  movable  arm  that  carries  the  vernier  V.     The 


THE   SEXTANT. 


295 


arm  is  centered  under  the  mirror.  This  mirror  is  called  the 
index  glass.  The  vernier  reads  to  ten  seconds.  H  is  another 
glass,  called  the  horizon  glass.  Its  lower  portion  is  a  mirror, 
and  its  upper  portion  is  unsilvered.  CrCr  are  colored  glasses 
to  protect  the  eye  when  making  observations  on  the  sun.  They 
may  be  turned  back  out  of  the  way  when  not  needed. 

278.   Use.     To  use  the  sextant  to  measure  an  angle  between 
two  terrestrial  objects,  hold  the  plane  of  the  arc  in  the  plane  of 


the  observer's  eye  and  the  two  points.  The  telescope  should  be 
directed  toward  the  fainter  object.  It  may  be  necessary  to  hold 
the  sextant  upside  down  to  do  this.  Swing  the  vernier  arm  till 
the  image  of  the  second  point  reflected  from  I  to  IT  to  the 
telescope  is  seen  superimposed  on  the  fainter  object  seen 
directly.  Clamp  the  vernier,  bring  the  images  into  exact 
coincidence,  using  the  tangent  screws,  and  read  the  vernier. 
The  reading  is  the  angle  sought.  It  will  be  observed  that  the 
angle  read  is  riot  horizontal,  unless  the  distant  points  are  in 
the  same  level  with  the  observer.  It  requires  some  practice 


296 


HYDROGRAPHIC   SURVEYING. 


to  become  expert  in  the  use  of  the  sextant.  The  eyepieces 
shown  in  the  figure  are  of  different  kinds.  There  is  usually 
one  astronomical  eyepiece,  one  for  terrestrial  work,  and  one 
without  magnifying  power. 

The  instrument  is  that  used  by  seamen  for  observing  for 
latitude  and  longitude. 

279.    Theory.     The  principle  on  which  the  sextant  is  con- 
structed is  as  follows :   If  a  ray  of  light  is  reflected  successively 


FIG.  144. 


from  two  plane  mirrors,  the  angle  between  the  incident  and 
finally  reflected  ray  is  twice  the  angle  of  the  mirrors. 

Referring  to  Fig.   144,  since  the  angles  of  incidence  and 
reflection  are  equal,  i  =  r  and  i'  =  r\  and  from  Geometry 


and 


V 


(i  +  r}-  (i1  +  /)  =  2  (r  -  r'), 
(90°  -  i')  -  (90°  -  r)  -  (r  -  r'). 


Therefore  E  =  2  I77,  which  was  to  be  shown. 

To  show  that  the  angle  read  by  the  vernier  on  the  arc  or 
limb  is  the  angle  F7,  suppose  the  two  mirrors  to  be  parallel  and 
the  telescope  to  be  directed  to  an  object  infinitely  distant,  so 


THE   SEXTANT.  297 

that  the  rays  from  it  are  parallel.  An  image  of  the  object  will 
be  seen  in  the  same  line  as  the  object,  or  will  appear  superim- 
posed on  the  object.  This  will  be  evident  from  the  equality  of 
the  angles  a.  The  vernier  will  then  be  in  the  position  V  and 
this  point  of  the  arc  is  graduated  zero.  If  now  the  vernier 
is  moved  to  another  point  V  for  the  purpose  of  getting  the 
reflection  of  a  second  point  to  cover  the  direct  image  of  the 
first  point,  the  angle  VIV  will  equal  that  at  F7,  the  angle 
of  the  mirrors.  It  is,  however,  the  angle  at  E  that  is  required, 
and  this  is  twice  the  angle  through  which  the  vernier  has 
moved.  Hence  the  arc  FT7,  instead  of  being  numbered  to 
give  the  angle  V,  is  numbered  to  give  at  once  the  angle  E. 
That  is,  each  degree  is  marked  two  degrees,  etc. 

280.  Adjustments.  There  are  four  adjustments  of  the  sextant: 

(1)  To  make  the  index  glass  perpendicular  to  the  plane  of 
the  limb. 

(2)  To  make  the  horizon  glass  perpendicular  to  the  limb. 

(3)  To  make  the  line  of  collimation  of  the  telescope  parallel 
to  the  plane  of  the  limb. 

(4)  To  make  the  vernier  read  zero  when  the  mirrors  are 
parallel. 

(1)  Bring  the  vernier  to  about  the  middle  of  the  arc.    Hold 
the  eye  at  about  the  upper  Gr  of  Fig.  143,  and  observe  the  arc 
near  the  zero  point  directly,  and  the  reflected  image  of  the 
other  end  of  the  arc  in  the  index  glass.     If  the  glass  is  perpen- 
dicular to  the  plane  of  the  limb,  the  reflected  and  direct  por- 
tions will  seem  to  form  one  continuous  arc.     Adjust  the  glass, 
if  necessary,  by  means  of  the  screws  at  its  base.     It  may  be 
necessary  to  place  slips  of  thin  paper  under  the  base. 

(2)  Direct  the  telescope  toward  a  star  or  other  very  distant 
object  and  note  whether  the  direct  and  reflected  images  seem, 
when  the  vernier  is  moved,  to  separate  or  overlap  laterally. 
If  they  do  not,  the  horizon  glass  is  in  adjustment.     The  adjust- 
ment is  made,  when  necessary,  by  the  screws  of  the  glass. 

(3)  Place  the  sextant  on  a  plane  surface  and  direct  the  tel- 
escope to  a  point  not  far  away.      Place  the  two  peep  sights 
shown  in  Fig.  143  on  the  extreme  ends  of  the  limb  and  note 
whether  the  line  of  sight  through  them  coincides  in  elevation 


298 


HYDROGRAPHIC    SURVEYING. 


with  the  line  of  sight  of  the  telescope.     If  not,  adjust  the  tele- 
scope by  the  screws  in  its  collar.     Any  other  objects  of  equal 

altitude  will  serve 
as  well  as  the  peep 
sights. 

(4)  Bring  the  di- 
rect and  reflected 
images  of  a  very  dis- 
tant point  to  coincide, 
and  read  the  vernier. 
It  should  read  zero. 
If  it  does  not,  note  the 
error  as  an  index  cor- 
rection to  be  applied 
to  all  angle  readings. 
The  error  may  be  cor- 
rected by  adjusting  the 
horizon  glass,  but  this 
is  not  usually  done. 


Fro.  145. 


281.  Other  forms. 
By  a  double  sextant, 
Fig.  145,  two  angles  can  be  measured  quickly  by  one  observer, 
one  sextant  measuring  one  angle  and  the  other  sextant,  the  other. 


MEASURING  VELOCITY  AND  DISCHARGE. 

282.  Position  of  maximum  velocity.      The   velocity   of   a 
stream  varies  in  different  portions  of  a  cross  section  and  in 
different  cross  sections.     The  maximum  in  a  cross  section  is, 
when  there  is  no  wind,  at  about  one  third  the  depth  in  the 
middle   of  the  channel.     The  surface  velocity  may  be  greater 
if  the  wind  is  favorable.     To  determine  the  mean  velocity  in  a 
given  cross  section,  the  velocity  in  many  parts  of  the  section 
must  be  found.     This  is  best  done  with  a  current  meter. 

283.  Current  meters.     These  are  of  various  patterns.    Those 
shown  in  Figs.  146-148  are  considered  good  forms.     Fig.  146 
is  a  meter  devised   by  W.   G.  Price,  United   States  assistant 
engineer.      There  is  an  electric  connection  with  a  register  that 


300  HYDROGRAPHIC   SURVEYING. 

indicates  the  number  of  revolutions  of  the  wheel.  The  number 
of  revolutions  per  second  is  a  function  of  the  velocity  of  the 
current  through  the  meter. 

Fig.  147  is  a  modification  of  the  Price  meter,  called  an 
audible,  or  acoustic,  meter,  and  for  streams  is  probably  as  con- 
venient as  any  form  yet  devised.  It  is  so  constructed  that  at 
each  tenth  or  twentieth  revolution  a  small  gong  is  struck,  and 
the  sound  is  carried  through  a  rubber  tube  to  the  ear  of  the 
observer  above  in  a  boat.  This  meter  requires  no  registering 
apparatus  at  all.  It  is  not  recommended  for  deep-sea  work, 
in  which  a  strong,  heavy  instrument  is  needed,  since  it  is  very 
light. 

Fig.  148  is  a  different  form  of  meter  with  electric  register- 
ing attachment. 

284.  Use  of  the  meter.     In  use  the  meter  is  simply  held  in 
different  positions  in  a  cross  section  whose  velocity  is  required, 
and  in  each  position  the  number  of  revolutions  per  second  is 
noted.     The  position  of  the  meter  is  located  by  any  one  of 
many  possible  methods   that  will  suggest   themselves   to   the 
surveyor.     The  number  of  revolutions  per  second  bears  some 
relation   to   the   velocity   of   the   current   going   through   the 
propeller-like  wheel  of  the  meter,  and  the  determination  of  this 
relation  is  called  "rating  the  meter."     Every  meter  must  be 
rated   before    the    velocity   of    current    corresponding    to    an 
observed  velocity  of  wheel  can  be  told. 

285.  Rating  the  meter.     The  meter  is  rated  by  moving  it 
through  still  water  at  a  known  rate  and  noting  the  revolutions 
per  second.     By  moving  it  at  different  velocities  it  will  become 
apparent  that  the  velocity  is  not  strictly  proportional  to  the 
speed  of  the  wheel  but  bears  a  relation  that  must  be  expressed 
by  an  equation.     From  a  number  of  observations  at  different 
velocities,  a  diagram  may  be  drawn  that  will  give  the  velocity 
corresponding  to  any  observed  speed  of  the  wheel.      This  is 
done  as  follows:     The  meter  is  moved  over  a  known  distance 
and  a  stop  watch  is  used  to  determine  the  time  occupied.     The 
distance  divided  by  the  time  in  seconds  gives  the  velocity  in 
feet  per  second.     The  number  of  revolutions  made  during  the 
time  is  recorded,  and  this  divided  by  the  time  gives  the  number 


MEASURING   VELOCITY  AND   DISCHARGE.  301 

of  revolutions  per  second.     The  following  observations  were 
made  by  Mr.  Price,  all  on  a  distance  of  two  hundred  feet. 

In  the  table,  R  is  the  whole  number  of  revolutions  in  the 
two  hundred  feet,  T  the  whole  time,  r  the  revolutions  per 
second,  and  v  the  velocity  per  second. 

No.  £  T          r          v 

1  100  53  1.886  3.774 

2  101  44  2.295  4.544 

3  101  41  2.464  4.878 

4  96  124  0.774  1.613 

5  94  152  0.618  1.316 

6  90  193  0.466  1.036 

7  91  181  0.503  1.105 

8  103       28      3.678      7.142 

9  100       53       1.886       3.774 

10  98       73       1.342       2.740 

11  193       29       3.552 


11)19.464 38.818 

1.769 


The  average  speed  of  the  wheel  is  seen  to  be  1.769  revolu- 
tions per  second,  and  the  average  velocity  of  flow,  or  current, 
is  3.529  feet  per  second. 

Two  axes  at  right  angles  are  now  drawn.  On  one  of  these 
axes  are  laid  off  the  several  values  of  r,  and  parallel  to  the 
other  are  laid  off  the  corresponding  values  of  v.  In  Fig. 
149,  the  horizontal  axis  is  that  of  r,  and  the  vertical  axis  is  the 
velocity  axis.  In  the  first  observation,  r  is  1.886  and  v  is  3.774. 
Assuming  suitable  scales  the  observations  are  plotted  in  the 
same  way  as  a  point  by  latitudes  and  longitudes.  Thus,  the 
latitude  of  the  first  observation  is  3.774  and  its  longitude  is 
1.886.  If  cross-section  paper  is  used  the  work  will  be  facili- 
tated. Each  observation  being  plotted,  it  is  observed  that  they 
all  fall  nearly  on  a  straight  line.  It  may  be  assumed  that  they 
should  all  fall  on  the  straight  line  and  that  they  do  not  because 
of  small  errors  in  the  observations.  If  the  mean  values  of  r  and 
v  are  plotted  it  may  be  assumed  that  the  line  should  pass  through 
the  point  thus  found  and  that  it  must  then  be  swung  to  average 
the  other  points  as  nearly  as  may  be.  A  piece  of  fine  thread  is 
stretched  through  the  point  of  average  r  and  u,  and  swung 


302 


HYDROGRAPHIC   SURVEYING. 


around  till  it  seems  to  the  eye  to  average  the  other  points. 
This  line  is  that  on  which  it  is  most  probable  that  the  observa- 


Axis  of  Revolutions  per  Second. 
FIG.  149. 

tions  should  all  fall,  and  hence  if  it  is  drawn  and  the  meter  is 
held  in  a  running  stream  and  the  revolutions  per  second  are 


MEASURING  VELOCITY   AND   DISCHARGE.  303 

noted,  the  corresponding  velocity  may  be  determined  by  laying 
off  the  observed  r  on  its  axis  and  measuring  the  corresponding 
v  up  to  the  line.  If  a  horizontal  line  is  drawn  on  the  diagram 
through  &,  where  the  line  cuts  the  axis  of  v,  it  will  be  seen  that 
for  any  value  of  r  the  corresponding  value  of  v  is  r  tan  a  +  bo. 
If  b  represents  60,  we  may  write 

v  =  r  tan  a  +  b. 
From  the  values  of  the  average  r  and  v  in  the  example  given 

v  =  1.904  r  +  0.16. 
In  its  general  form  this  equation  is 

v  =  ar  +  5, 

and  this  is  known  in  Analytical  Geometry  as  an  equation  of  a 
straight  line.  This  equation  is  the  general  equation  of  the  rela- 
tion between  v  and  r.  The  rating  of  the  meter  then  consists 
in  finding  values  for  a  and  b  from  observations  that  give  v  and 
r.  Since  there  are  two  unknown  quantities,  two  independent 
observations  should  be  sufficient ;  but  since  no  observation  can 
be  perfect,  many  are  taken,  and  mean  values  are  determined  for 
a  and  b.  This  may  be  done  analytically  by  the  method  of  Least 
Squares. 

286.  Rod  floats.  A  very  good  method  of  obtaining  veloci- 
ties, when  a  current  meter  is  not  available,  is  to  observe  the 
velocities  of  rod  floats  in  various  parts  of  the  stream.  For  this 
purpose  two  ranges  are  established  on  shore,  one  above  and  one 
below  the  section  in  which  it  is  desired  to  measure  the  velocity. 
The  ranges  are  laid  out  normal  to  the  stream.  A  transit  is 
placed  on  each  range,  and  the  rod  is  started  just  above  the 
upper  range.  The  transit  man  on  the  upper  range  signals  the 
lower  observer  when  the  rod  is  about  to  cross  the  upper  range, 
and  just  as  it  crosses ;  and  the  lower  observer  reads  an  angle  to 
the  rod  at  the  instant  it  crosses  the  upper  range.  The  opera- 
tion is  repeated  when  the  rod  crosses  the  lower  range,  the  lower 
observer  making  the  signals,  and  the  upper  observer  reading 
the  angle  to  the  rod.  Each  notes  the  time  of  crossing  the 
range  on  which  he  is  stationed.  In  this  way  the  path  of 
the  rod  and  the  time  it  takes  to  travel  that  path  are  known. 
The  rod  may  be  of  wood,  or  it  may  be  a  long  cylindrical  tin 
can.  In  either  case  it  should  be  weighted  so  as  to  remain 


304  HYDROGRAPHIC   SURVEYING. 

vertical  and  just  clear  the  bottom.  The  velocity  of  the  float 
will  then  be  approximately  that  of  the  vertical  filament  of  the 
stream  in  which  it  floated.  The  immersion  of  the  rod  should 
be  about  nine  tenths  of  the  depth  of  the  stream.  If  this  rule  is 
observed,  the  mean  velocity  in  the  vertical  filament  will  be  given 
by  Vm  =  V0  [1  -  0.116  (VI)  -  O.I)]1  in  which  V0  is  the  observed 
velocity  of  the  rod,  and  D  is  the  ratio  of  the  depth  of  the  water 
below  the  bottom  of  the  rod  to  the  total  depth  of  the  water. 

In  small  streams,  wires  or  ropes  graduated  with  tags  may 
be  stretched  across  the  stream  at  the  upper  and  lower  ends  of 
the  stretch  of  river  to  be  observed,  and  the  time  of  passing  of 
the  float  under  each  wire  or  rope  may  be  observed. 

By  observing  the  velocities  in  many  vertical  filaments,  the 
mean  velocity  of  the  section  may  be  determined.  It  would  not 
be  a  mean  of  the  several  observed  velocities  unless  all  "fila- 
ments "  had  equal  areas.  The  mean  velocity  is  that  which  multi- 

plied by  the  area  will  give  the  discharge,  hence  it  is  -  -r  -  —  • 


287.  The  discharge.     To  get  the  discharge,  simply  multiply 
the  mean  velocity  of  each  filament  by  its  cross-sectional  area, 
and  add  the  products.     If  the  velocity  is  determined  by  meter, 
the  work  need  be  done  in  but  one  cross  section  ;    if  by  rod 
floats,  it  will  usually  be  best  to  measure  two  or  more  sections, 
preferably  at  the  quarter  points  of  the  stretch,  from  which  to 
get  a  mean  section. 

DIRECTION  OP  CURRENT. 

288.  How  determined.     It  is  sometimes  necessary  to  know 
the  direction  of  both  surface  and  subsurface  currents.     The 
necessity  may  arise  in  the  determination  of  the  proper  place  to 
discharge  sewage,  or  in  surveys  for  the  improvement  of  har- 
bors and  the  approaches  thereto.     The  surface  currents  may  be 
determined  by  watching  floats.     The  subsurface  currents  are 
best  found  by  a  direction  meter.     This  is  a  form  of  meter  that 
not  only  gives  velocities,  but  also  shows  the  direction  of  the 
current  in  which  it  is  placed.      It  was  devised   by  Messrs. 
Ritchie  and  Haskell.     It  is  said  to  be  one  of  the  best  forms  of 
meters  for  measuring  currents  alone. 

1  Francis's  "  Lowell  Hydraulics." 


CHAPTER   XII. 

MINE  SURVEYING. 
SURFACE  SURVEYS. 

289.  Coal  mines.  The  surface  surveys  in  connection  with 
coal  mining  operations  consist  in  making  land  surveys  of  'the 
property  that  may  be  owned  by  the  mining  company,  and  locat- 
ing all  shafts,  tunnel  openings,  buildings,  railways,  roads,  and 
other  structures.  In  addition  to  this  it  is  well,  if  the  area 
owned  is  extensive,  to  make  a  complete  topographical  survey 
of  the  property.  Frequently  a  mining  company  mines  coal 
under  the  lands  of  many  small  holders  and  pays  a  royalty  for 
the  coal  so  mined.  It  is  therefore  necessary  to  know  the  posi- 
tions of  the  land  lines  of  all  owners  under  whose  land  it  is 
probable  that  operations  may  extend.  The  location  of  these 
lines  is  best  performed  by  careful  stadia  surveys  —  reading  the 
stadia  to  the  smallest  possible  unit.  (There  may  be  cases  of 
sufficient  importance  to  warrant  most  careful  transit  and  tape 
work.)  The  topographical  survey  may  be  carried  on  at  the 
same  time.  Indeed,  the  location  of  the  land  lines  becomes 
merely  an  incident  in  the  topographical  survey.  The  positions 
of  the  corners  are  determined  by  computing  their  latitudes  and 
longitudes. 

Among  other  points  that  will  be  located  in  the  topograph- 
ical survey  or  land  survey  will  be  the  positions  of  exploration 
drill  holes  that  have  been  put  down  in  advance  of  mining  oper- 
ations for  the  purpose  of  locating  veins,  or  beds,  as  they  are 
frequently  called.  The  entire  survey  may  or  may  not  be  hung 
on  a  system  of  triangles.  If  it  is  of  any  considerable  extent, 
it  is  best  to  reference  it  thus  for  the  sake  of  the  checks  it 
gives  on  the  field  work,  even  though  not  more  than  two  or 

B'M'U  SCRV.—  W  305 


306  MINE  SURVEYING. 

three  triangles  are  used.  The  whole  work,  except  the  side 
shots,  is  best  plotted  by  latitudes  and  longitudes,  these  being 
computed  with  sufficient  accuracy  for  this  purpose  by  the  aid 
of  a  trigonometer.  The  side  shots,  except  those  to  land  corners, 
may  be  plotted  with  a  protractor.  The  surface  map  should 
show  all  outcroppings  of  mineral. 

290.  Metal  mines.     Surface  surveys  for  metal  mines  may  be 
of  the  same  character  as  those  for  coal  mines,  but  in  the  United 
States  there  are  certain  laws  governing  the  size  and  form  of 
mining  claims  that  are  located  on  public  land,  and  the  method 
of  surveying  them. 

291.  Form  of  mining  claim.     By  the  provisions  of   these 
laws,  a  mining  claim  may  not  exceed  fifteen  hundred  feet  in 
length  along  the  vein  nor  six  hundred  feet  in  width  (three 
hundred  feet  each  side  of  the  vein).     It' may  be  of  any  length 
or  width  less  than  these  limits,  according  to  local  custom  ;  but 
no  rule  may  be  made  that  will  compel  the  claim  to  be  less  than 
twenty-five  feet  on  each  side  of  the  vein.     The  side  lines  need 
not  be  straight  or  parallel,  but  the  end  lines  must  be  both. 
The  miner  has  the  right  to  follow  the  vein  he  has  discovered 
into  the  ground  to  any  depth,  even  should  it  depart  so  much 
from  the  vertical  as  to  pass  beyond  the'vertical  planes  through 
the  side  lines ;  but  he  may  not  mine  beyond  the  vertical  planes 
through  the  end  lines. 

When  claims  intersect,  the  subsequently  located  claim  is 
said  to  conflict  with  the  previous  one  and  carries  with  it  only 
the  area  that  is  not  in  conflict.  The  foregoing  rules  apply 
to  what  are  called  lode  claims. 

A  "  lode  "  is  the  name  sometimes  given  to  a  vein  of  ore. 

A  "  placer  "  claim  is  a  claim  taken  for  the  purpose  of  ob- 
taining gold  or  silver  or  other  metal  from  the  surface  materials, 
as  in  washing  gold  from  a  river  bed  or  from  gravel  in  the  mines 
worked  by  hydraulic  process  in  California.  Such  a  claim  may  not 
exceed  twenty  acres,  and,  if  located  on  surveyed  lands,  its  lines 
must  conform  to  the  legal  subdivisions.  A  placer  claim  carries 
no  right  to  mine  beyond  the  vertical  planes  through  its  bound- 
ing lines.  It  carries  the  right  to  mine  a  vein  discovered  within 


SURFACE   SURVEYS.  307 

its  boundaries  after  the  claim  is  located,  but  not  if  the  vein  is 
known  to  exist  when  the  claim  is  located,  unless  specially 
mentioned. 

292.  Surveying  the  claim.     When  a  miner  has  found  a  vein 
of  ore,  he  has  a  claim  staked  out  on  the  ground,  conforming  to 
the  legal  requirements,  using  a  width  and  length  that  he  thinks 
best.     He  also  posts  the  notices  required  by  law.     This  survey 
may  be  made  by  any  surveyor.     After  the  miner  has  expended 
five  hundred  dollars  in  working  his  claim  he  is  entitled  to 
have  -it   deeded    to   him   by  the  government.      This  deed  is 
called  a  "patent,"  and  the  claim  is  said  to  be   "patented." 
For  the  purpose  of  this  patent,  a  survey  is  made,  from  which 
to  write  a  description.     This  final  survey,  on  which  the  deed 
to  the  miner  is  based,  must  be  made  by  a  United  States  offi- 
cer, known  as  a  United  States  deputy  mineral  surveyor.     Such 
officers  are  under  bonds  of  ten  thousand  dollars,  and  are  re- 
quired to  pass  an  examination  before  receiving  their  licenses. 
They  make  the   final   survey"  and  tie  it  to  a  corner   of   the 
public  surveys  or  to  a  special  mineral  survey  monument,  and 
return  to  the  surveyor  general  of   the  state  the  notes,  with 
an  affidavit  that  the  required  five  hundred  dollars  have  been 
expended  on  the  mine. 

The  form  of  field  notes  and  the  instructions  issued  to  United 
States  deputy  mineral  surveyors  by  the  surveyor  generals  of  the 
various  mining  states  may  be  had  on  application  to  the  surveyor 
general's  office. 

293.  Surface   monuments.      The  surface   monuments   of   a 
mine  that  is  to  be  worked  for  any  length  of  time  should  be  of 
a   permanent   character.      At   least   two   of    them   should   be 
located  near  the  entrance  to  the  mine,  and  these  two  should 
be  particularly  permanent,  for  use  in  connecting  the  under- 
ground and  surface  surveys.     These  may  be  auxiliary  monu- 
ments located  for  the  express  purpose  of  connecting  the  two 
surveys.     They  may  be  built  of  stone  or  brick,  with  a  properly 
centered  copper  bolt  to  mark  the  exact  spot.     The  bolt  may 
be  removable  for  the  purpose  of  substituting  a  signal.     These 
monuments  must  be  constructed  with  a  view  to  being  proof 


308  MINE   SURVEYING. 

against  disturbance  by  frost,  or  travel,  or  by  the  operations 
about  the  mine.  A  true  north  and  south  line  permanently 
monumented  is  perhaps  the  best  base  for  all  surveys. 


UNDERGROUND   SURVEYS. 

294.  General   statement.     Underground  surveys  consist  of 
traversing,  leveling,  and  sometimes  measuring  volumes.     The 
necessary  traverses  are  run  to  determine  the  location  of   the 
various  portions  of  the  mine,  so  that  a  map  or  a  plan  of  the 
mine  can  be  made.     Leveling  is  done  to  determine  the  relative 
elevations  of  the  different  parts  of  the  mine,  the  grade  of  the 
tunnels  or  drifts,  the  direction  and  amount  of  dip  of  the  vein, 
etc.     The  quantity  of  ore  mined  is  sometimes  determined  by 
measuring  the  volume  excavated  in  the  mine,  and  the  meas- 
urement of  the  quantity  of  ore   "  in  sight "  is  usually  by  the 
volumetric  method.     The  ore  mined  is  usually  measured  by 
weight  after  it  is  brought  out  of  the  mine. 

The  only  differences  between  underground  and  surface 
surveys  are  due  to  the  difficulties  encountered  in  working  in 
dark,  cramped  passages.  These  necessitate  certain  modifica- 
tions in  surface  methods  to  make  them  applicable  to  under- 
ground work. 

295.  Definitions.     The   following  are   a  few  definitions  of 
technical  terms  used  in  mining.1     There  are  many  others  that 
the  mining  surveyor  must  know. 

Shaft.     A  pit  sunk  from  the  surface. 

Adit.  A  nearly  horizontal  passage  from  the  surface,  by 
which  a  mine  is  entered  and  through  which  the  water  that 
collects  in  it  is  removed.  In  the  United  States  an  adit  is 
usually  called  a  tunnel,  though  the  latter,  strictly  speaking, 
passes  entirely  through  a  hill,  and  is  open  at  both  ends. 

Drift.  A  horizontal  passage  underground.  A  drift  follows 
the  vein,  as  distinguished  from  a  cross-cut,  which  intersects  it, 
or  a  level  or  gallery,  which  may  do  either. 

Level.     A  horizontal  passage  or  drift  into  or  in  a  mine.     It 

1  Taken  from  "  A  Glossary  of  Mining  and  Metallurgical  Terms  "  by  R.  W.  Ray- 
mond, mining  engineer. 


UNDERGROUND  SURVEYS.  309 

is  customary  to  work  mines  by  levels  at  regular  intervals  in 
depth,  numbered  in  their  order  below  the  adit  or  drainage  level. 

Cross-cut.     A  level  driven  across  the  course  of  a  vein. 

Winze.     An  interior  shaft,  usually  connecting  two  levels. 

Strike.  The  direction  *  of  a  horizontal  line,  drawn  in  the 
middle  plane  of  a  vein  or  stratum,  not  horizontal. 

Dip.  The  inclination  of  a  vein  or  stratum  below  the  hori- 
zontal. The  dip  is  necessarily  at  right  angles  2  with  the  strike 
or  course,  and  its  inclination  is  steeper  than  that  of  any  other 
line  drawn  in  the  plane  of  the  vein  or  stratum. 

Pitch.  The  inclination  of  a  vein,  or  of  the  longer  axis  of 
an  ore  body. 

Incline  or  Slope.  A  shaft  not  vertical ;  usually  on  the  dip 
of  a  vein. 

Stope.  To  excavate  ore  in  a  vein  by  driving  horizontally 
upon  it  a  series  of  workings,  one  immediately  over  the  other, 
or  vice  versa.  Each  horizontal  working  is  called  a  stope  (prob- 
ably a  corruption  of  step),  because  when  a  number  of  them  are 
in  progress,  each  working  face  being  a  little  in  advance  of 
the  next  above  or  below,  the  whole  face  under  attack  as- 
sumes the  shape  of  a  flight  of  steps.  The  process  is  called 
overhand  stoping,  when  the  first  stope  is  begun  at  a  lower 
corner  of  the  body  of  ore  to  be  moved,  and,  after  it  has  ad- 
vanced a  convenient  distance,  the  next  is  commenced  above  it, 
and  so  on.  When  the  first  stope  begins  at  an  upper  corner, 
and  the  succeeding  ones  are  below  it,  it  is  underhand  stoping. 
The  term  "  stoping  "  is  loosely  applied  to  any  subterranean  ex- 
traction of  ore  except  that  which  is  incidentally  performed  in 
sinking  shafts,  driving  levels,  etc.,  for  the  opening  of  the  mine. 

296.  Location  and  form  of  station  marks.  In  underground 
work,  stations  can  not  be  stakes  driven  in  the  floor  of  the  mine. 
The  reasons  are  obvious.  Where  the  roof  is  solid  rock,  holes 
may  be  drilled  in  it  and  wooden  plugs  inserted,  in  which  may  be 
driven  a  nail  or  tack.  A  good  form  is  a  horseshoe  nail  with 
flattened  head,  in  which  is  cut  a  triangular  hole  with  an  apex  of 
the  triangle  toward  the  head  end  of  the  nail.  Such  a  nail 

1  Azimuth. 

2  That  is,  the  direction  of  the  vertical  plane  in  which  the  amount  of  the  dip  is 
measured  is  at  right  angles. 


310  MINE   SURVEYING. 

driven  in  the  roof  insures  that  a  plumb  line  suspended  from  it 
will  always  hang  from  the  same  point.  A  small  round  opening 
about  one  eighth  inch  in  diameter  is  preferred  by  some.  When 
the  roof  is  soft  and  the  drift  timbered,  temporary  stations  may 
consist  of  nails  driven  in  the  timbers.  It  is  probably  better  in 
such  cases  to  reference  the  station  by  two  points  on  the  side  of 
the  drift.  If  the  two  points  are  properly  located,  and  the  dis- 
tance from  the  vertical  through  the  station  to  each  of  them  is 
recorded,  the  station  may  be  recovered  at  any  time. 

When  about  to  occupy  a  station,  transfer  it  to  the  floor 
with  a  plumb  line  and  set  there  a  temporary  mark  consisting 
of  a  piece  of  slate  with  a  cross  scratched  on  it,  or  a  small 
metallic  cone  carried  for  the  purpose.  Stations  underground 
should  be  plainly  marked.  White  paint  is  good  if  kept  bright. 
The  stations  in  a  mine  should  be  consecutively  numbered,  and 
the  same  number  should  not  be  used  twice.  They  should  be 
numbered  at  the  station  points  as  well  as  in  the  notebook. 
Probably  as  good  a  way  to  do  this  as  any  is  to  drive  round- 
headed  nails  or  tacks  in  the  timbering  near  the  station,  locating 
them  all  in  the  same  position  relative  to  the  station.  The  tacks 
may  be  driven  in  a  small  board  or  plug  fastened  to  the  side 
of  the  drift  near  the  station.  The  nails  are  arranged  to  give 
the  number  of  the  station,  thus  •::  for  324  and  »0:  for  302, 

a  washer  being  used  for  the  0.  Practically  this  method  was 
followed  in  the  New  Almaden  mine  in  California.  The  nail 
heads  can  be  felt  and  read,  even  though  not  clearly  visible. 

As  the  mine  is  extended  and  new  stations  are  added,  they 
should  receive  their  proper  numbers  and  be  permanently  lo- 
cated. There  is  no  branch  of  surveying  where  method  and 
system  count  for  so  much  as  in  mine  surveys. 

297.  Instruments  used.  For  all  angle  work  in  mine  surveys, 
except  the  location  of  short  tortuous  passages,  the  transit  is 
the  only  suitable  instrument.  The  work  must -be  done  with 
great  precision,  llesults  need  not  be  so  precise,  perhaps,  as  in 
high  class  city  work,  but  a  precision  of  one  in  five  thousand 
should  certainly  be  secured.  It  is  best  to  use  azimuths  and 
perform  the  traverses  as  already  described  under  traversing 
with  the  transit. 


UNDERGROUND   SURVEYS.  311 

The  transit  should  be  a  complete  transit  or  tachymeter,  so 
that  leveling  may  be  done  either  directly  or  by  vertical  angles. 
The  vertical  angle  method  is  the  better  ordinarily,  as  it  saves 
much  time  in  the  mine.  The  tripod  should  be  either  short  or 
adjustable  in  length.  A  diagonal  eyepiece  is  desirable  for 
pointings  at  high  inclinations.  The  wires  are  made  visible  by 
means  of  a  reflector  described  in  Art.  90,  or  by  a  hollow  axis 
and  reflector  within  the  telescope.  Often  enough  light  is 
secured  by  having  an  assistant  flash  a  candle  near  the  objective. 
It  is  sometimes  necessary  to  point  upward  or  downward  at  a 
considerable  inclination,  as  when  an  incline  is  to  be  traversed. 
Pointing  up  is  accomplished  by  means  of  an  auxiliary  diagonal 
eyepiece,  but  pointing  down  can  not  be  done  when  the  angle  is 
so  great  that  the  line  of  sight  cuts  the  plates  of  the  instrument. 

Various  devices  have  been  invented  to  overcome  this  diffi- 
culty. Sometimes  an  aux- 
iliary telescope  is  mounted 
above,  and  parallel  with,  the 
main  telescope  of  the  tran- 
sit, as  in  Fig.  150.  In 
some  transits,  the  auxiliary 
telescope  is  mounted  on  the 
end  of  the  horizontal  axis, 
as  in  Fig.  151.  Some  tran- 
sits are  made  with  the 
standards  leaning  forward 
far  enough  to  permit  the 
line  of  sight  to  clear  the  FlG-  150' 

plates  when  turned  vertical.  Others  are  made  with  double 
standards,  one  as  usual,  and  one  bracketed  to  the  ordinary 
standards.  The  telescope  is  made  so  as  to  be  readily  changed 
from  one  bearing  to  the  other.  If  the  side  telescope  is  used, 
the  transit  should  be  set  over  a  point  to  one  side  of  the  station 
occupied,  or  the  line  of  collimation  of  the  auxiliary  telescope 
should  be  directed  to  a  point  as  much  to  one  side  of  the  true 
station  as  the  auxiliary  telescope  is  to  one  side  of  the  main 
telescope.  This  is  conveniently  done  by  having  the  object 
sighted  to,  as  a  candle,  fastened  to  a  flag  at  the  proper  distance 
from  it.  The  flag  is  held  on  the  true  station. 


UNDERGROUND   SURVEYS.  313 

For  tortuous  passages  in  which  a  transit  can  not  be  set  up, 
so  crooked  that  one  end  can  not  be  seen  from  the  other, 
resort  is  had  to  what  is  sometimes  called  a  German  dial,  and  a 
hanging  clinometer.  One  of  the  best  forms  of  the  dial  is  shown 
in  Fig.  152.1 

The  compass  is  hung  on  a  wire  or  line  stretched  from  one 
station  to  the  next,  and  as  it  levels  itself  like  a  mariner's  com- 
pass, it  may  be  properly  read.  The  bear- 
ings are  not  depended  on,  but  angles  are 
read.  Passages  so  tortuous  as  to  require 
the  use  of  such  methods  are  not  usually 
of  great  length,  and  the  resulting  errors 
are  small.  They  are  not  likely  to  occur 
in  coal  mines,  but  occur  frequently  in 
quicksilver  mines,  and  sometimes  in  gold 
and  silver  mines.  Coal  mines  usually  FlG-  152- 

involve  fewer  irregularities  of '•alignment  than  any  other  class 
of  mines. 

The  linear  measurements  are  made  with  a  steel  tape.  In 
some  cases  of  long,  straight  drifts,  a  long  tape,  two  hundred 
feet,  or  three  hundred  feet,  may  be  used  to  advantage ;  but 
usually  a  tape  either  fifty  feet  or  one  hundred  feet  long  is  best. 
The  tape  should  be  as  long  as  possible,  and  no  more  stations 
should  be  used  than  seem  necessary.  It  is  very  necessary  to 
expedite  work  in  a  mine,  because  mining  work  in  a  drift  must 
be  largely  suspended  while  the  survey  is  in  progress.  The 
flat  wire  tapes  are  best  for  mine  work,  since  they  are  not  so 
easily  broken  as  the  ribbon  tapes.  A  pocket  tape,  graduated  to 
hundredths  of  feet,  may  be  carried  for  the  necessary  plus  meas- 
urements at  stations. 

If  a  level  rod  is  used,  it  must  be  a  short  rod.  Bench  marks 
are  best  placed  on  the  side  of  the  drifts  where  they  will  not  be 
disturbed. 

298.  Devices  for  making  stations  visible.  The  object  ob- 
served at  a  station  is  usually  a  plumb  line  made  visible  by 
holding  a  piece  of  oiled  paper  or  milk  glass  behind  it  and  a 
candle  behind  the  transparency.  The  plumb  line  may  be  illu- 

i  From  Brough's  "  Treatise  on  Mine  Surveying." 


814 


MINE   SURVEYING. 


minated  by  holding  a  light  in  front  of  it,  shielded  from  the 
observer. 

Another  method  is  to  sight  the  flame  of  a  candle  or  lamp, 
properly  centered  over  the  station.     Another  way  is  to  use 

a  plummet  lamp,  which  is  a 
lamp  made  in  the  form  of  a 
plummet.  Perhaps  the  neatest 
method,  at  least  in  coal  mines, 
is  to  use  a  second  and  third 
tripod  on  which  can  be  mounted 
a  target  lamp,  such  as  is  shown 
in  Fig.  153.  The  lamp  is  be- 
hind the  target  from  the  ob- 
server, and  the  target  may  be 
accurately  centered  over  the 
station  point.  The  target  and 
transit  are  interchangeable  on 
the  tripods,  the  particular  form 
shown  being  used  with  a  special 
transit  that  is  interchangeable 
above  the  leveling  screws. 
The  method  of  use  is  some- 
what as  follows  :  The  back 
station  is  occupied  by  a  target, 
as  is  also  the  forward  station, 
and  that  at  which  the  obser- 
vation is  to  be  made  is  occu- 
pied by  the  transit.  When 
the  observation  is  complete, 
the  tripod  at  the  back  station 
is  carried  to  the  next  forward 
station,  the  others  remaining 
in  place.  The  back  lamp 
is  placed  on  the  tripod  that 
was  occupied  by  the  transit 
while  the  latter  is  being  carried  to  the  tripod  occupied  by 
the  previous  forward  lamp,  which  in  turn  is  carried  to  the 
next  forward  station.  The  lamps  should  neither  of  them  be 
moved  till  the  observations  at  the  station  occupied  by  the 


FIG.  153. 


UNDERGROUND  SURVEYS.  315 

transit  are  complete.  This  is  a  very  expeditious  method 
of  work.1 

The  entire  angular  work  may  be  done  first,  and  the  linear 
measurements  made  afterward,  or  vice  versa,  or  tho  two  may  be 
carried  on  at  the  same  time.  The  latter  method  will  probably 
consume  more  of  the  miner's  time,  since  the  linear  work  would 
retard  the  angle  work.  The  linear  work  may  be  carried  on 
without  seriously  interfering  with  the  work  of  the  mine.  The 
extent  or  character  of  the  work  Avill  guide  the  surveyor  in 
choosing  his  methods.  When  lateral  drifts  depart  at  frequent 
intervals  from  one  long  tunnel  or  drift,  the  stations  at  the 
angle  points  of  the  main  drift  should  be  first  located  and  the 
substations  to  be  used  in  running  the  lateral  drifts  should  be 
set  in  the  line  of  the  main  drift.  This  may  be  done  at  the 
time  the  angles  of  the  main  drift  are  measured.  In  this  case, 
the  angle  work  should  be  done  first. 

When  an  extensive  survey  of  a  mine  is  to  be  made,  the  sur- 
veyor should  go  through  the  mine  and  lay  out  his  programme 
of  work  as  closely  as  may  be.  It  will  almost  always  be  possible 
to  mark  out  the  main  stations  before  taking  the  transit  into  the 
mine,  or  at  least  before  setting  it  up. 

299.  Notes.  In  an  underground  traverse  in  which  notes 
are  to  be  taken  at  intervals  along  each  course  to  objects  on 
the  sides,  a  good  form  is  as  follows :  Prepare  the  page  of  the 
notebook  by  ruling  a  column  down  the  center,  in  which  to 
write  distances  and  azimuths,  and  leave  the  sides  to  note  the 
objects  measured  to,  and  such  other  information  as  need  be 
recorded.  The  form  would  be  similar  to  that  shown  on  page 
229,  except  that  the  column  for  distances  would  also  contain 
the  azimuths  and  would  be  in  the  middle  of  the  page  instead 
of  at  one  side.  Notes  taken  on  the  right  of  a  line  that  is  run 
out  should  be  noted  on  the  right  of  the  column,  and  those 
taken  on  the  left  should  be  recorded  on  the  left. 

1  An  important  point  in  favor  of  the  three-tripod  system,  when  long  tapes  are 
used  for  measuring,  is  that  all  ordinary  distances  may  be  measured  from  center  of  the 
telescope  to  center  of  back-sight  or  fore-sight  flame,  the  vertical  angle  being  read  and 
noted.  Having  then  the  length  of  the  line  of  sight  and  its  angle,  the  true  horizontal 
distance  between  stations  is  easily  calculated.  Oftentimes  this  is  a  much  safer  course 
than  to  rely  on  presumably  horizontal  measurements  along  the  floor  of  the  gangways. 

J.  J.  ORMSBEE,  Mining  Engineer. 


316  MINE   SURVEYING. 

A  line  should  be  drawn  across  the  page  between  each  two 
courses. 

For  the  purpose  of  computation  of  latitudes  and  longitudes, 
the  azimuths  and  distances  may  be  copied  in  more  compact 
form  in  the  office  notebook. 

If  elevations  are  carried  by  vertical  angles,  a  good  form 
is  that  given  in  Art.  127  for  traversing  with  the  transit  and 
stadia,  with  sufficient  vertical  space  given  each  course  to  write 
the  details  of  the  line  along  the  side  or  right-hand  page  under 
remarks.  The  center  line  up  the  middle  of  the  right-hand 
page  may  be  used  to  represent  the  transit  line,  and  the  objects 
may  be  sketched  to  scale  on  this  page.  There  should  always 
be  two  sets  of  notes  in  existence,  kept  in  different  places,  and 
when  not  in  use  they  should  be  stored  in  fire-proof  vaults. 


CONNECTING  SURFACE   AND  UNDERGROUND  WORK. 

300.  When  the  mine  is  entered  by  a  tunnel.     In  order  that 
the  underground  workings  may  be  plotted  with  reference  to  the 
surface  lines,  or  even  so  that  they  may  be  properly  oriented,  it 
is  necessary  to  connect  the  underground  surveys  with   those 
on  the  surface.      The  simplest  case  occurs  when  the  mine  is 
entered  by  an  adit  or  drift.     From  a  point  in  the  surface  sur- 
vey, located  conveniently  for  the  purpose,  run  a  traverse  into 
the  drift, 

301.  When  there  are  two  shafts.     If  the  mine  is  entered  by 
shafts,  and  there  are  two  or  more  of  these  some  distance  apart, 
the  surface  and  underground  surveys  are  connected  by  running 
a  traverse  on  the  surface  and  another  underground  between  the 
shafts.     The  azimuth  of  the  line  joining  the  shafts  is  computed 
from  the  surface  survey  ;  and  that  of  the  same  line  underground, 
from  the  underground  survey,  a  zero  azimuth  being  assumed  for 
this  purpose.  The  two  computed  azimuths  will  differ  by  the  angle 
between  the  true  zero  and  the  assumed  zero,  and  the  assumed 
underground  azimuths  may  be  corrected  accordingly.    The  pre- 
cise points  at  the  shafts  are  made  the  same  by  using  plumb 
lines.     In  deep  shafts  the  lines  are  best  made  of  piano  wire, 
and  the  plummet  must  w.eigh  from  ten  to  twenty-five  pounds. 


CONNECTING  SURFACE  AND  UNDERGROUND  WORK.  317 

The  line  is  first  lowered  with  a  light  bob,  and  the  heavy  one 
is  attached  at  the  bottom.  The  line  may  be  lowered  by  a 
reel.  Signals  may  be  given  for  lowering  or  raising  by  tapping 
on  the  guides  or  pipes  in  the  shaft.  Wherever  possible,  this  is 
an  excellent  method,  since  small  errors  of  position  of  the  bobs 
are  not  multiplied. 

Another  method,  that  may  be  applied  in  shallow  mines,  is 
to  use  the  transit  and  auxiliary  telescope,  either  setting  the 
points  in  the  bottom  in  line  with  a  given  line  on  top,  or 
setting  the  transit  in  the  bottom  by  trial,  so  that  the  line  of 
collimation  may  be  in  line  with  a  surface  line.  Perhaps  a 
better  way  than  the  latter  is  to  set  the  transit  on  the  bottom 
and,  with  the  horizontal  motion  clamped,  set  two  points  in  a 
line  on  opposite  sides  of  the  shaft  at  the  surface,  which  line 
is  produced  into  the  mine  as  the  first  course  of  the  traverse. 
The  points  located  on  the  surface  are  afterward  connected 
with  the  surface  survey.  This  method  of  using  the  transit 
and  auxiliary  telescope  is  adapted  only  to  shallow  shafts  that  are 
tolerably  free  from  smoke.  The  better  method  is  that  using 
the  plummets.  Work  may  be  carried  from  one  level  to  another 
through  winzes  and  air  shafts  by  the  same  method.  As  high 
vertical  angles  occur  in  this  work,  double  centering  is  an  abso- 
lute essential  to  correct  work. 

Levels  are  carried  down  the  shafts  by  measuring  with  a 
steel  tape,  or,  what  is  the  same  thing,  by  lowering  a  weighted 
wire  and  then  measuring  the  wire.  If  the  tape  is  used, 
and  is  shorter  than  the  shaft,  it  is  first  hung  on  a  nail  near 
the  surface,  the  elevation  of  the  nail  being  determined  by 
leveling.  The  reel  and  observers  are  then  slowly  lowered 
in  the  cage  until  the  tape  is  all  paid  out,  when  a  second  nail 
is  driven  at  a  measured  distance  from  the  first.  A  piece  of 
white  cloth  is  fastened  to  this  second  nail,  to  make  it  visible ; 
and  the  cage  is  raised,  the  tape  unhooked,  the  cage  again 
lowered  to  the  second  nail,  and  the  operation  repeated  from 
there  down.  A  nail  is  left  for  a  bench  mark  opposite  each 
level. 

302.  When  the  mine  is  entered  by  one  shaft.  To  connect 
a  surface  survey  with  an  underground  survey  through  a  single 


318 


MINE   SURVEYING. 


deep  shaft  is  the  most  difficult  task  the  mining  surveyor  will 
encounter.  If  the  mine  is  of  small  extent  and  the  shaft  shal- 
low, it  may  be  done  with  the  transit  and  auxiliary  telescope, 
as  already  described  under  the  preceding  article.  But  if  the 
shaft  is  deep,  it  can  not  be  done  this  way.  The  method  is  to 
use  two  plumb  lines.  The  utmost  care  is  necessary  in  their 
use,  for  from  a  base  line  from  six  to  twelve  feet  long  must  be 
produced  a  survey  one  or  two  or  more  miles  long.  It  may  be 
necessary  to  drive  a  second  shaft  a  mile  or  two  distant,  and  it 
must  be  driven  to  meet  an  underground  drift.  An  error  of 
one  tenth  of  an  inch  in  a  base  line  ten  feet  long  will  cause  a 
shaft  two  miles  distant  to  be  driven  nearly  nine  feet  out  of 
place.  Two  methods  of  using  two  plumb  lines  have  given 
satisfactory  results. 

(1)  In  Fig.  154,  M  and  M'  are  surface  monuments,  and  AB 

the  two  plumb 
lines.  The  dis- 
tance AB  is  care- 
fully measured, 
and  the  distance 
MA  and  MB  and 
the  angles  at  M. 
The  azimuth  of 
AB  is  then  deter- 
mined. T  is  the 
transit  set  in  the 
mine  below.  The 
angles  at  T,  and 
the  distances  AT 
and  BT  are  meas- 
ured. The  triangle  TAB  is  then  solved,  from  which  the  azi- 
muth of  TD  becomes  known. 

(2)  This  method  is  probably  not  so  good  as  the  second, 
which   consists   in   suspending   the   plumb   lines   so   that   the 
transit  may  be  set  up  in  line  with  them  below.      The  diffi- 
culty in  all  this  work  is  that  the  plumb  lines  are  never  still, 
but  continually  oscillate.       The  mean  position  of   the  line  is 
best  determined  by  placing  a  fine  scale  behind  it  and  noting 
the  amplitude  of  the  vibrations.     The  transit  is  pointed  to  the 


FIG.  154. 


CONNECTING  SURFACE  AND  UNDERGROUND  WORK.  319 

mean  position.  The  oscillations  are  not  in  a  straight  line,  but 
are  more  or  less  elliptical;  hence  the  scale  must  be ^placed  a 
little  behind  the  wire,  but  should  be  as  close  as  possible  to  it, 
in  order  to  avoid  parallax.  The  transit  is  set  up  in  the  mine 
as  close  to  the  nearest  line  as  possible  (this  will  be  from  ten 
to  fifteen  feet  distant),  and  each  line  is  separately  observed 
and  the  transit  gradually  brought  by  trial  into  line  with  the 
mean  positions  of  both  wires.  The  two  plummets  should  hang 
in  water,  or,  better,  in  some  more  viscous  liquid,  as  oil,  or  even 
molasses.  The  oscillations  will  be  slow,  a  line  three  hundred 
and  twenty-five  feet  long  requiring  about  ten  seconds  for  one 
complete  vibration  in  air.  Of  course  the  cages  can  not  be  in 
use  during  the  operation. 

The  points  from  which  the  bobs  are  suspended  must  be 
as  firm  as  possible  and  as  well  defined,  so  that  the  line  join- 
ing them  may  be  connected  with  the  surface  surveys.  Two 
transits  are  really  required  to  insure  good  work.  One  is 
left  on  the  surface  with  an  observer  to  see  that  there  is  no 
movement  of  the  points  of  suspension  during  the  operation, 
and  to  note  what  -change,  if  any,  takes  place.  It  must  be 
known  that  the  lines  do  not  hang  against  any  projections  on 
the  sides  of  the  shaft.  This  may  be  told,  in  some  instances, 
by  looking  up  the  shaft ;  in  others,  by  passing  a  candle  or  other 
light  slowly  around  the  wire  at  the  bottom,  and  observing  from 
the  top  that  it  is  visible  throughout  the  motion.  In  case 
neither  method  is  practicable,  a  system  of  movements  of  the 
wire  may  be  arranged.  When  the  observer  below  is  ready,  he 
will  signal  by  rapping  on  the  pipes  or  guides,  and  the  observer 
above  will,  at  agreed  intervals  of  time,  move  the  wire  out  one 
inch,  two  inches,  etc.  The  observer  below  notes  whether  cor- 
responding movements  take  place  there. 

On  account  of  the  stoppage  of  machinery  this  connection 
of  surface  and  underground  surveys  is  an  expensive  operation. 
It  should  nevertheless,  in  important  cases,  be  performed  several 
times,  and  a  mean  of  the  results  taken  for  the  first  course  of 
the  underground  traverse.  When  the  bottom  of  the  shaft  can 
be  seen  from  the  top,  a  method  used  at  the  Severn  tunnel  will 
give  good  results.  A  wire  is  stretched  from  one  side  of  the 
shaft,  about  one  hundred  or  more  feet  into  the  drift,  and  accu- 


320  MINE  SURVEYING. 

rately  aligned  for  the  few  feet  visible  by  a  transit  on  the  sur- 
face. In  the  case  of  the  Severn  tunnel  the  wire  was  stretched 
over  horizontal  screws,  the  wire  lying  in  the  groove  between 
the  threads.  The  wire  was  then  moved  laterally  for  aligning 
by  turning  the  screws. 

MAPPING   THE  SURVEY. 

303.  Metal  mines.      Three  maps  are  necessary  for  a  com- 
plete representation  of   a  mine  with   many  levels  running  in 
different   directions  —  (1)   the   plan ;  (2)  a  longitudinal   sec- 
tion ;   (3)  a  transverse  section.     Such  a  set  of  maps  is  shown 
in  Plate  IV.  at  the  back  of  the  book.     The  notes  taken  by  the 
•surveyor  should  be  such  as  to  enable  him  to  make  these  three 
maps.     These  maps  may  be  in  addition  to  one  of  the  surface 
survey,  and,  if  so,  there  would  be  four  maps  necessary  for  a 
complete  representation  of  a  mining  property.     It  is  usual  to 
tint  the  portions  shown  on  the  plans  as  worked,  and  the  differ- 
ent levels  are  sometimes  tinted  with  different  colors  to  distin- 
guish them.     If  several  lodes  are  shown  on  one  map,  it  will  be 
better  to  color  each  lode,  working  with  a  single  color  to  dis- 
tinguish it  from  the  other  lodes. 

304.  Coal   mines.      The  representation  of   a  coal  mine  is 
usually  a  simpler  matter  than  the  mapping  of  a  metal  mine. 
The  coal  ordinarily  lies  in  beds  nearly  horizontal,  and  the  work- 
ings may  often  be  represented  by  a  plan  alone,  the  different 
levels  being  distinguished  by  colors,  and  the  elevations  being 
written  in.     It  is  customary  to  show  on  the  map  the  direction 
and  amount  of  dip  of  the  seams.     There  are  a  number  of  sym- 
bols used  to  represent  various  objects ;  but  the  practice  is  not 
uniform.     Arrows  may  be  used  to  show  the  direction  of  air 
currents,  blue  for  inlets,  and  red  for  outlets,  etc.     The  plan 
should  show  all  the  details  of  the  mine,  all  stop  walls,  all  doors, 
all  sumps  or  reservoirs,  all  pumps  or  machinery,  the  location  of 
faults,  etc. 

It  is  extremely  necessary  that  complete,  correct  maps  should 
be  kept  up,  so  that  if  any  connections  are  to  be  made,  they  may 
be  correctly  directed.  Very  serious  accidents  have  occurred 
from  incorrect  maps  of  mines. 


MAPPING  THE  SURVEY.  321 

305.  Scale.     The  state  of  Pennsylvania  provides  that  there 
shall  be  kept  at  each  mine  a  plan  on  a  scale  of  one  to  twelve 
hundred,  or  one  hundred  feet  per  inch,  for  the  use  of  the  state 
inspector.     The  English  law  requires  the  plan  to  be  on  a  scale 
of  twenty-five  inches  per  mile,  or  about  one  half  that  required 
in    this   country.      Other   countries   require    different    scales. 
The  maps  of  the  mining  claims  made  by  the  deputy  mineral 
surveyors  are  to  a  scale  of  two  hundred  feet  to  the  inch.     This 
would  seem  to  be  large  enough  for  working  drawings,  except  for 
details  of  structures. 

As  in  the  case  of  notes,  two  maps  should  be  in  existence 
and  kept  in  different  places.  A  good  example  of  a  coal  mine 
map  is  shown  in  Plate  V.  at  the  back  of  the  book. 

306.  Problems.     The  problems  that  arise  in  mining  survey- 
ing  call   for  some  ingenuity  in  handling   Trigonometry   and 
Descriptive  Geometry,  although  they  may  all  be  solved  without 
a  knowledge  of  the  latter  subject  by  name. 

Some  problems,  such  as  seeking  a  lost  vein,  require  more  the 
principles  of  Geology  than  those  of  Mathematics. 
The  commonest  problems  are  the  following  : 

(1)  The  determination  of  the  course  and  distance  from  a 
point  in  one  compartment  of  a  mine  to  a  point  in  another  com- 
partment.     By  course  is  meant  azimuth  and  vertical  angle. 
This  includes  such  problems  as  the  determination  of  the  dis- 
tance and  direction  to  run  a  tunnel  to  tap  a  shaft. 

(2)  Knowing  the  strike  and  dip  of  a  vein,  to  determine  the 
length  of  a  shaft  or  drift  started  at  a  given  point  and  run  in  a 
given  direction  to  meet  the  vein,  or  to  determine  the  direction 
to  run  a  tunnel  from  a  given  point  so  as  to  cut  the  vein  in  the 
shortest  possible  distance,  and  to  find  the  distance  required. 

It  is  believed  that  the  student  can  solve  such  problems  as 
are  given  in  the  Appendix,  page  340,  if  he  remembers  that  the 
pitch  or  dip  is  always  measured  in  a  vertical  plane  normal  to  a 
horizontal  element  of  the  vein,  and  that  the  strike  is  the  direc- 
tion of  such  horizontal  element. 
R'M'D  SURV.  — 21 


APPENDIX. 

I.    PROBLEMS   AND   EXAMPLES. 
CHAPTER   T. 

THE  problems  in  the  use  of  the  chain  may  be  solved  on  the  blackboard 
with  string  and  chalk. 

1.  To  range  out  a  line  between  two  points  on  opposite  sides  of  a  hill : 
Mark  the  points  so  that  they  will  be  visible  from  the  top  of  the  hill.     Two 
men  with   range   poles  then   place   themselves  near  the  top  of  the  hill, 
facing  each  other,  so  that  each  can  see  one  end  of  the  line.     They  then 
alternately  align  each  other  with  the  visible  ends  of  the  line  until  both 
are  in  the  line.     The  same  principle  is  used  in  "fudging"  one's  self  into 
line  between  two  distant  visible  points. 

2.  To  range  a  line  across  a  deep  ravine  or  valley  :    The  inability  to  do 
this  by  eye  alone  arises  from  the  fact  that  the  eye  can  not  carry  a  vertical 
line  on  a  hillside,  nor  can  it  be  sure  of  transferring  a  point  vertically  down- 
ward, even  in  level  country.     The  observer  stands  on  one  side  of  the  valley, 
swinging  a  plummet  over  a  point  in  the  line  to  be  ranged.      He  places 
his  eye  in  line  with  the  plumb  line  and  a  distant  point  in  the  line  to  be 
ranged,  and,  casting  his  eye  down  the  plumb  line,  directs  the  placing  of  a 
flag  held  by  an  assistant  in  the  valley. 

3.  To  erect  a  perpendicular  to  a  line  :    A  triangle  whose  sides  are  in  the 
ratio  4,  5,  6,  is  a  right  triangle.     Hence,  fasten  one  end  and  the  thirty-link 
division  at  one  point  in  the  line  and  the  ten-link  division  at  another  point. 
Carry  the  eighteen-link  division  out  till  the  three  portions  of  the  chain 
used  form  a  triangle.     If  the  line  is  to  be  erected  at  a  given  point,  the 
method  will  suggest  itself  to  the  student. 

4.  Using  the  chain  and  pins  in  the  way  a  string  and  pencil  or  a  pair  of 
dividers  would  be  used  on  the  drawing  board,  perform  the  above  problem  in 
a  different  way. 

5.  To  drop   a  perpendicular  on   a  line  from  a  given  point :   Run  an 
inclined  line  from  the  point  to  the  given  line,  erect  a  perpendicular  to  the 
latter,  and  produce  it  till  it  intersects  the  inclined  line.      The   required 


PROBLEMS   AND   EXAMPLES.  323 

perpendicular  is  parallel  to  the  perpendicular  erected,  and  the  consideration 
of  the  similar  figures  will  enable  the  student  to  solve. 

6.  To  run  a  line  parallel  to  a  given  line  :  Erect  two  equal  perpendiculars 
to  the  given  line.     If  the  parallel  is  to  pass  through  a  given  point,  drop  a 
perpendicular  from  the  point  to  the  line  and  erect  a  perpendicular  to  this  at 
the  point.     Otherwise  :  The  diagonals  of  a  parallelogram  bisect  each  other. 

7.  To  measure  an  angle  with  a  chain  :   In  an  isosceles  triangle  of  legs  «, 
included  angle  A,  and  base  b,  2  a  sin  \  A  =  b. 

8.  To  lay  out  a  given  angle  :  Reverse  the  process  just  suggested. 

9.  A  line  is  measured  with  a  chain  that  is  afterward  found  to  be  one 
link  too  long,  and  is  found  to  be  10.36  chains  long.     What  is  its  true 
length  ? 

10.  A  line  is  measured  with  a  100-foot  tape  and  found  to  be  723.36  feet 
long.     The  tape  is  afterward  found  to  be  0.02  foot  short.     What  is  the  true 
length  of  the  line? 

11.  A  triangular  field  is  measured  with  a  chain  that  is  afterward  found 
to  be  one  link  too  long.     The  sides  as  measured  are  0  chains,  4  chains,  and 
3  chains,  respectively.     What  is  the  resulting  area  and  what  is  the  true 
area? 

12.  An  irregular  field  is  measured  with  a  chain  three  links  short.     The 
area  is  found  to  be  36.472  acres.     What  is  the  true  area? 

13.  A  100-foot  tape  is  standard  length  for  a  pull  of  10  pounds,  when 
supported  its  entire  length,  and  at  a  temperature  of  62°  F.      A  line  is 
measured  and  found  to  be  1000.00  feet  long.     The  tape  is  of  steel,  has  a 
cross  section  of  0.002  square  inch,  and  weighs  0.00052  Ib.  per  inch.     In 
the  measurement  it  is  at  a  temperature  of  80°  F.,  and  is  unsupported  except 
at  the  ends.     The  pull  is  25  pounds.     "What  is  the  true  length  of  the  line  ? 

14.  A  steel  tape  weighs  0.0061  pounds  per  lineal  foot.     It  is  100  feet  long 
and  is  standard  for  no  pull  when  entirely  supported,  at  62°  F.     It  has  a 
cross  section   of  0.002  square  inch.     What  pull  must  be  exerted  to  keep 
the  tape  standard  length  if  it  is  supported  only  at  its  ends,  the  temperature 
remaining  unchanged? 

15.  A  line  is  measured  along  the  surface  on  a  hillside.     The  first  300 
feet  has  a  vertical  angle  of  3°,  the  next  250  feet  has  a  vertical  angle  of  4°  30', 
and  the  last  700  feet  has  a  vertical  angle  of  1°.     What  is  the  true  length  of 
the  line  ? 

16.  A  similar  line,  of  equal  surface  lengths,  is  measured,  and  the  rise 
per  hundred  feet  of  each  stretch  is  found  to  be  respectively  5  feet,  8  feet, 
and  If  feet.     What  is  the  true  length  of  the  line? 

17.  Measure  a  line  of  about  1000  feet  or  more  in  length,  at  least  twice 
with  a  chain  and  twice  with  a  tape,  and  determine  the  difference  of  measure- 
ment.   Do  this,  if  possible,  over  two  lines  of  about  equal  length,  but  offering 
very  unequal  difficulties  to  measurement,  and  note  the  differences. 


324  APPENDIX. 


CHAPTER  III. 

1.  Determine  the  angular  value  of  one  division  of  each  bubble  avail- 
able in  the  school  collection  of  levels.    The  method  is  in  brief  as  follows : 
Hold  a  rod  at  a  known  distance ;  read  the  rod  and  bubble  with  the  bubble 
near  one  end  of  its  tube,  and  then  read  with  the  bubble  near  the  other 
end.     By  the  differences  of  readings  and  the  known  distances,  determine 
the  value  of  one  division.     If  there  are  any  striding  levels,  use  these  as 
described  for  the  level  with  metal  base. 

2.  Other  exercises  with  the  level  will  suggest  themselves  to  the  student 
or  teacher.     A  considerable   amount  of    differential   and  profile  leveling 
should  be  done,  and   profiles   drawn.      The   leveling   should    always    be 
checked  by  rerunning  in  the  opposite  direction. 


CHAPTER   IV. 

1 .  Radiating  from  a  point  A ,  are  eight  lines  of  bearings  as  follows : 

AB,  N.  53°  45'  W.  AF,  S.  86°  45'  E. 

A  C,  N.  36°  42'  W.  A  G,  S.  24°  36'  E. 

AD,  N.  18°  34'  E.  A H,  S.  20°  30'  W. 

AE,  N.  34°  28'  E.  A  I,   S.  36°  20'  W. 

Required  the  angles  between  the  following  bracketed  lines  —  always 
considering  the  smaller  angle  : 

\AB  ^(AB  (AB             (AB             (AB             (AB            (AB 

\AC  \AD  \AE            (AF            (AG            (AH            (AI 

(AC  (AC  ^AC             (AC             (AC            (AC 

(AD  \AE  (AF            (AG            I  AH            (  AI 

(AD  (AD  (AD            (AD            (AD 

'(AE  (AF  (AG             (AH            ( AI 

(AE  (AE  (AE            (AE 

(AF  (AG  (AH            (AI 

(AF  (AF  (AF 

(AG  (AH  (AI 

(AG  (AG 

\AH  (AI 

(AH 

(AI 

2.  A  traverse  is  run  with  the  following  bearings  :   N.  42°  E.,  N.  36°  E., 
S.  1°  W.,  N.  50°  W.     Determine  the  interior  angles.    If  the  traverse  is  of 
a  closed  field,  what  should  be  the  sum  of  the  interior  angles? 


PROBLEMS   AND   EXAMPLES.  325 

3.  A  traverse  is  run  with  the  following  azimuths,  zero  azimuth  being 
north :  42°,  144°,  181°,  230°.     What  are  the  bearings? 

4.  Another  is  run  with  the  following  azimuths :  306°  15',  18°  34',  93°  15', 
200°  30'.     What  are  the  bearings? 

5.  What  are  the  azimuths  of  the  following  lines  of  Example  1 :  AC, 
AE,AG,AI1 

6.  Determine  the  exterior  or  deflection  angles  in  Example  4. 

7.  Let  the  eight  lines  of  Example  1  be  the  first  eight  consecutive  courses 
of  a  traverse.     Determine  the  deflection  angles. 

8.  The  magnetic  declination  at  a  given  place  is  found  to  be  10°  30'  E. 
What  will  be  the  bearing  of  the  true  north  ?  the  true  south  ?  the  true  east  ? 
the  true  west? 

9.  The  magnetic  declination  is  7°  10'  W.     What  will  be  the  bearings  of 
the  four  true  cardinal  points  ? 

10.  A  line  of  an  old  survey  is  recorded  as  N.  18°  E.  mag.  bear.     It  now 
reads  N.  16°  30'  E.     What  has  been  the  change  in  declination  in  direction 
and  amount  ? 

11.  A  line  of  an  old  survey  is  recorded  as  N.  36°  15'  E.  mag.  bear.     It 
now  reads  N.  34°  30'  E.,  and  the  magnetic  declination  is  now  10°  30'  W. 
What  was  the  declination  at  the  time  of  the  original  survey? 

12.  A  line  of  an  old  survey  is  recorded  as  N.  36°  15'  E.  mag.  bear.,  and 
the  declination  is  recorded  as  having  been  10°  30'  W.  at  the  time  of  the  sur- 
vey.    The  declination  is  now  12°  00'  W.     What  magnetic  bearing  should 
the  line  now  show  ? 

13.  A  line  of  an  old  survey  is  recorded  as  S.  26°  00'  E.  mag.  bear.,  dec- 
lination 10°  30'  W.     The  declination  is  now  9°  W.     What  magnetic  bear- 
ing should  be  used  to  retrace  the  line? 

14.  The  magnetic  declination  is  10°  30'  E.     If  the  declination  vernier  is 
attached  to  the  south  side  of  the  compass  box,  in  what  direction  and  by 
what  amount  should  it  be  moved  so  that  true  bearings  may  be  read  by  the 
needle  ?     With  sights  pointed  to  the  magnetic  north,  what  would  the  needle 
read  after  moving  the  vernier  ? 

15.  A  line  of  an  old  survey  is  recorded  as  N.  30°  15'  E.  mag.  bear.     The 
declination  is  now  4°  W.,  and  the  same  line  reads  N.  30°  E.     It  is  desired 
to  set  the  declination  vernier  so  that  the  remainder  of  the  survey  may  be 
retraced  by  the  recorded  bearings.     The  vernier  being  attached  to  the  south 
side  of  the  compass  box,  what  is  its  movement  in  amount  and  direction? 

16.  Suppose,  in  the  above  example,  the  former  bearing  had  been  S.  26° 
W.,  and  the  present  bearing  S.  26°  30'  W.     What  would  be  the  movement 
of  the  vernier? 

17.  Is  it  necessary  in  the  foregoing  examples  to  know  the  present  decli- 
nation ? 

18.  Determine  the  angular  value  of  the  plate  bubbles  and  the  telescope 
bubble  of  each  transit  in  the  school  collection.     Do  this  work  as  suggested 


826 


APPENDIX. 


for  the  level,  and  also  for  the  telescope  bubble,  by  the  use  of  the  vertical 
circle. 

19.  Determine  the  meridian  by  an  observation  on  Polaris  and  by  the 
solar  transit,  and  compare  the  results. 

20.  Measure  all  the  angles  of  a  polygon  that  has  been  laid  out  on  the 
ground  and  note  whether  they  sum  up  properly. 

21.  Set  over  each  station  in  succession  and  run  the  polygon  as  a  traverse, 
using  azimuths,  and,  reoccupying  the  first  station,  redetermine  the  azimuth 
of  the  first  course  from  that  of  the  last,  and  note  whether  the  azimuth  found 
agrees  with  that  first  used. 

In  fairly  good  transit  work,  the  error  in  a  perimeter  of  a  mile  or  more 
should  not  exceed  one  minute. 

22.  The  adjustments  of  the  transit  should  be  made  by  the  student. 


CHAPTER  V. 

1.   Determine  for  the  transits  in  the  school  collection  the  value  of  4  and 


2.    Reduce  the  horizontal  distances  and  differences  of  elevation  in  the 
set  of  notes  shown  on  pages  337-338. 


CHAPTER   VI. 

Find  the  error  of   closure,  balance  the  survey,  plot,  and  compute  the 
areas  of  the  following  fields : 


STA. 

BEARING. 

CHAINS. 

A 

S.  51°  10'  E. 

5.05 

B 

S.  58°  10'  W. 

4.63 

C 

X.  29°  15'  W. 

4.16 

D' 

X.  45°  30'  E. 

2.87 

Area,  1.676  +  A. 

A 

X.  84°  00'  W. 

9.04 

B 

S.  21°  15'  W. 

12.34 

C 

X.  72°  15'  E. 

12.92 

I) 

X.    9°  30'  E. 

6.68 

S.  7°  25'  W. 
S.  62°  05'  W 
X.  2°  35'  E. 
X.  34°  25'  E. 
X.  83°  00'  E. 


Area,  9.264 


4.35 
6.94 
4.01 
3.64 
4.51 


Area,  3.124  +  A. 


PROBLEMS    AND   EXAMPLES. 


327 


4. 


STA. 

BEARING. 

CHAINS. 

1 

S.  62°  15'  E. 

5.12 

2 

S.  71°  15'  E. 

4.66 

3 

S.     5°  30'  W. 

12.00 

4 

S.  80°  15'  W. 

12.46 

5 

N.    2°  15'  E. 

9.46 

6 

N.  25°  45'  E. 

9.40 

Area,  17.683  +  A. 

STA. 

AZIMUTH. 

CHAINS  . 

1 

107°  15' 

16.40 

2 

196°  15' 

24.10 

3 

246°  05' 

19.60 

4 

13°  55' 

37.00 

Area,  48.36  +  A. 

C).    The  latitudes  and  longitudes  of  a  series  of  points  are  as  follows, 
measured  in  chains: 

(  Latitude  +  13.63 
1  Longitude  +  10.20 
5  Latitude  +  1 .26 

Longitude  + 
( Latitude  - 
1  Longitude  + 


<  Latitude     — 
1  Longitude  — 


5.10 
3.30 
7.20 
3.30 


(  Latitude     +    7.10 
I  Longitude  -  10.20 

Find  the  area  included  by  the  broken  line  joining  the  points.    (Shift  the 
coordinate  axes  so  that  the  coordinates  are  all  positive.) 

7.    The  following  offsets  were  taken  on  the  sides  of  a  line  indicated,  at 
points  100  feet  apart.     Required  the  area  between  extreme  boundaries. 


L. 

137.4 

93.5 

49.3 

78.2 

102.0 

100.5 

96.6 

144.2 

111.9 

71.4 


DIST.  R. 

0  55.1 

100  76.5 

200  83.3 

300  79.9 

400  09.7 

500  59.5 

600  83.3 

700  70.5 

800  54.4 

900  101.2 

Area,  153,085  square  feet. 


328  APPENDIX. 

8.  The  following  offsets  were  taken  on  one  side  of  a  base  line,  to  deter- 
mine the  area  between  it  and  an  irregular  boundary  line.     The  distances 
are  all  from  zero : 

Distance*  Offset. 

Feet.  Feet. 

0  20.7 

202  31.5 

358  42.6 

825  53.2 

984  3G.1 

1223  40.7        Area,  49,698.25  sq.  ft. 

9.  In  Example  1,  let  it  be  supposed  that  the  second  course  is  wanting. 
Supply  it. 

10.  In  Example  2,  let  it  be  supposed  that  the  bearing  of  the  second 
course  and  the  length  of  the  third  course  are  wanting.     Supply  them. 

11.  In  the  third  example,  let  it  be  supposed  that  the  bearings  of  the  sec- 
ond and  fourth  courses  are  wanting.     Supply  them. 

12.  In  the  fifth  example,  let  it  be  assumed  that  the  lengths  of  the  second 
and  fourth  courses  are  wanting.     Supply  them. 

13.  The  wheel  of   a  planimeter  has   a   circumference  of  2.26  inches. 
What  should  be  the  length  of  tracer  that  the  number  of  revolutions  multi- 
plied by  10  shall  give  in  square  inches  the  area  circumscribed  ? 

14.  An  area   is  circumscribed    by  a   planimeter  with  the  fixed   point 
inside,  and  the  resulting  reading  is  found  to  be  11.26  square  inches.     The 
area  is  known  to  be  120  square  inches.     What  is  the  area  of  the  zero  cir- 
cumference ? 


COORDINATES.^ 
MODEL  EXAMPLE. 

1.  To  determine  the  coordinates  of  the  corners  Abcdef  (Fig.  155)  and 
plot  the  survey,  the  corner  e  and  bearing  fe  being  unknown,  but  point  e'  in 
prolongation  of  fe  and  bearing  of  de,  as  well  as  other  corners,  known  : 

Run  the  random  traverse  ABCDEF.  Make  a  rough  sketch  of  the 
traverse  and  place  on  it  lengths  of  lines  and  angles  measured.  (Note  that 
angles  and  not  azimuths  or  bearings  are  read.  This  is  careful  work  where 
angles  are  repeated  several  times,  and  hence  it  is  impracticable  to  read 
azimuths.)  Balance  the  angles  so  that  their  sum  will  be  "  twice  as  many  right 
angles  less  four  as  the  figure  has  sides,"  distributing  any  error  equally 
among  the  various  angles.  In  the  example  the  angles  "close." 

1  The  problems  here  given  were  furnished  the  author  by  Mr.  John  H.  Myers,  Jr., 
A.B.,  C.E.,  Assistant  Engineer  Department  of  Public  Works,  Brooklyn,  N.Y.  The 
examples  are  from  practice  in  New  York  City  and  were  treated  as  here  indicated. 


FIG.  155. 


330 


APPENDIX. 


Following  a  previous  survey  in  this  district,  assume  bearing  of  DE  to 
be  S.  9°  31'  E.,  and  compute  bearings  of  remaining  lines  and  place  them  on 
the  sketch. 

Assume  coordinates  of  point  A  200  N.  and  100  E.  to  bring  all  coordi- 
nates positive. 

Compute  the  coordinates  of  the  other  corners  of  the  traverse  as  shown 
on  page  331.  The  student  should  note  the  systematic  layout  of  the  work. 
The  error  of  closure,  0.05  E.  and  0.14  N.,  is  distributed  as  in  balancing  a 
survey,  and  the  balanced  coordinates  are  used  in  all  subsequent  work. 

Plot  the  traverse  as  shown  in  Fig.  155.  Check  the  plotting  by  scaling 
the  lengths  of  the  lines. 

To  locate  the  property  corners  ; 

Corner  A  is  a  corner  of  the  random. 

Course  .4/was  actually  run  on  the  ground. 

Calculate  from  the  recorded  angles  the  bearings  of  the  lines  joining  ran- 
dom corners  and  true  corners,  and  compute  the  latitude  and  longitude 
differences  of  these  lines,  and  from  these  the  coordinates  of  the  corners. 
They  will  be  found  to  be : 

b,  597.42  E.,  282.05  N. 

c,  585.10  E.,  353.95  N. 

d,  881.69  E.,  403.66  N. 
/,  118.59  E.,  70.33  N. 

The  computations  are  systematically  arranged  as  follows,  the  computa- 
tions for  c,  d,  and  /,  being  shown  : 


Dd 


N.  34°  32'  30"  W. 
29.81 

c  .  . 

S.  89°  51'  30"  W. 
11.49 

//  .  . 

9.753587 
1.474362 

602.00  E. 
16.90  W. 

9.915776 
1.474362 
1.390138 

7.417968 
1.060320 

329.40  N. 
24.55  N. 

1.227949 

9.999990 
1.060320 

585.10  E. 

893.18  E. 
11.49  W. 

353.95  N. 

403.69  N. 
0.03  S. 

1.060319 

881.69  E. 

403.66  X. 

S.    8°  09'  30"  E. 
131.00 

/    • 


9.152010 
2.117271 


100.00  E. 

18.59  E. 

118.59  E. 


9.995582 
2.117271 


200.00  N. 
129.67  S. 


By  Prob.  1,  Art.  143,  find  the  length  and  bearing  of  the  courses  the 
coordinates  of  whose  ends  are  known.  They  will  be  found  to  be  as  indi- 
cated on  the  figure.  The  arrangement  of  the  work  for  a  single  course  is 
shown  on  page  332. 


PROBLEMS   AND    EXAMPLES. 


331 


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ryi  M      fV,  rV] 

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sssi- 


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OS  <N 


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«3 


o  g. 
fc 


332 


APPENDIX. 


erf 

353.95 

403.66 

49.71 

1.696444 

2.472156 

9.224288 


585.10 


296.59 


N.  80°  29'  07"  E. 


tan 


53' 


2.478172 


i        do. 
300.73 


To  find  the  corner  e  find  the  coordinates  of  e',  then  the  bearing  of  e'f  by 
Prob.  I.,  Art.  143.  The  bearing  of  de  is  known  to  be  S.  9°  24'  30"  E.  By 
Prob.  II.,  Art.  143,  find  the  coordinates  of  e,  the  intersection  of  de  and  fe'. 
The  quantities  found  appear  on  the  sketch.  Prob.  II.  is  arranged  thus  : 


75°  48'  44" 
89°  37'  48" 


403.66  N. 
70.33  N. 

333.33 

2.522874 
2.882581 
9.640293 
9.962081 
2.920500 
9.986547 
0.000009 
2.907056 
807.34  ef 


881.69  E. 
118.59  E. 
763.10 


N.  66°  24'  14"  E. 
23°  35'  46" 


13°  49'  04" 


2.920500 
9.378097 
0.000009 
2.398606 
198.88  de 


From  the  length  and  bearing  of  de  and  coordinates  of  d  find  coordi- 
nates of  e,  and  from  coordinates  of  e  andy  find  bearing  of  fe. 

Plot  the  three  corners  as  the  randoms.  In  close  work  it  is  best  to  use 
seconds,  but  on  the  final  map  the  nearest  half  minute  may  be  recorded. 
Minutes  and  tenths  will  serve  as  well  as  minutes  and  seconds. 

If  tne  area  is  required,  it  is  obtained  as  follows,  making  the  computations 
by  logantnms.  The  work  is  here  carried  to  a  needless  number  of  decimals. 

00 

100.00 

100.00  x  282.05  =  28205.0000 
597.42  x  353.95  =  211456.8090 
585.10  x  403.66  =  236181.4660 
881.69  x  207.46  =  182915.4074 
914.20  x  70.33  =  64295.6860 
118.59  x  200.00  =  23718.0000 
746772.3684 


200.00\/282.05\/353.95\ /403.66\/207.46\/  70.33  \/200. 
lo6.JO/\597.42/\585.10/\881.69/\914.20/\118.59/\lOO. 


200.00  x  597.42  =  119484.0000 

282.05  x  585.10  =  165027.4550 

353.95  x  881.69  =  312074.1755 

403.66  x  914.20  =  369025.9720 

207.46  x  118.59  =  24602.6814 

70.33  x  100.00  =   7033.0000 

997247.2839 

746772.3684 

250474.9155  -=-  2 


125237.4577  square  feet  =  2.875  acres. 


PROBLEMS   AND   EXAMPLES. 


333 


EXAMPLES. 

1.  In  Fig.  155  determine  whether  the  line  fd  passes  to  the  north  or 
south  of  the  corner  b;  and  how  far  b  is  from  fd,  measured  on  a  line  at  right 
angles  tofd. 

2.  In  Fig.  156  the  line  ABCDEF  represents   a  portion  of  a  survey 
which  has  been  made  to  locate  a  street  or  railroad.     It  has  been  decided  to 
connect  the  lines  AB  and  £Fby  a  circular  curve  of  radius  716.779  feet,  and 
it  is  required  to  find  how  far  from  B  the  curve  will  start  and  how  far  from 
E  it  will  end.    The  method  in  outline  is  as  follows :  Assume  coordinates  for 


N.  597.83) 
E.  470.14  f 


FIG.  156. 


the  point  B,  and  by  means  of  the  connecting  courses  work  up  coordinates 
for  the  point  E.  By  Problem  I.,  Art.  143,  find  the  bearing  of  BE  and  the 
logarithm  of  its  length.  Then  by  Problem  II.  find  the  distances  from  B 
and  from  E  to  the  intersection  7  of  AB  and  EF  produced.  Then  the  radius 
multiplied  by  the  tangent  of  £  the  angle  GOH,  which  angle  is  equal  to 
the  angle  at  I,  gives  the  "tangent  distance"  HI,  and  this  subtracted 
from  IB  and  from  IE  gives  the  required  distances  GB  and  HE. 


334 


APPENDIX. 


3.  Fig.  157  illustrates  the  location  of  two  piers,  B  and  C,  for  the 
approach  to  a  bridge.  The  piers  were  on  a  curve  and  were  located  by 
the  usual  method  of  running  railway  curves,  the  necessary  deflections  and 

distances  being 
shown  in  the  fig- 
-X  ure.  As  the 
points  B  and  C 
would  have  to  be 
removed  during 


00. 
North 


i'oo'oo" 
'  io°oooo' 


FIG.  157. 


the  construction  of  the  piers,  it 
became  necessary  to  so  "reference  " 
them    that   they  could    easily   and 
quickly  be   recovered   at   any   time. 
This  was  done  by  placing  two  or  more 
points  on  the  radial  lines  through  the 
centers  of  the  piers,  since  the  intersec- 
tions of  these  lines  with  the   lines   AB 
and  AC  would  determine   the  points   B 
and  C.     If  the  ground  had  been  solid,  the 
instrument  could  have  been  set  at  B  and  at 
'    C,  and,  by  turning  off  the  proper  angles  from 
4,  the  radial  lines  BO  and  CO  could  easily  have 
been  established.     This,  however,  it  was  impos- 
sible to  do,  as  the  ground  was  so  swampy  as  to 
render  the  instrument  too  unsteady  for  good  work. 
The    following    method  was    therefore   adopted : 
Beginning  at  A,  a  random  traverse  Abcf  was  run 
on  firm  ground,  and  the  angles  and  distances  were 
measured  as  shown.     The  distances  from  b  and  from 
c  at  which  the  radial  lines  BO  and  CO  would  intersect 
this  traverse  were  then  calculated,  points  were  set  at  e 
and  d,  and  the  angles  beB  and  cdC  were  measured  to  see 
whether  their  actual  and  calculated  values  agreed.     This 
having  been  done,  the  instrument  was  set  at  e  and  at  rf, 
and  by  backsighting  on  B  and  C  and  reversing,  any  num- 
ber of  points  could  be  established  on  the  radial   lines  BO 
and  CO.     The  points  B  and  Care  thus  referenced.   The  prob- 
lem for  the  student  is  therefore  to  calculate  the  distances  be 
and  cd.     It  will  be  well  to  use  the  tangent  AX  as  a  meridian 
and  to  make  the  coordinates  of  A  00  N  and  00  E.     (Problem  II., 
Art.  113.) 


si 


PROBLEMS   AND   EXAMPLES. 


336 


FIG.  158. 


4.  From  the  data  given  on  Fig.  158 
calculate  the  depths  ac  and  brl  of  lot 
number  4. 

Center 
~  "Radius^oloO 


CHAPTER  VIII. 

1.  A  4°  curve  begins  at  station  36  -f  50, 
and  the  central  angle  is  16°  00'.     Tabulate 
the  deflection  angles  from  the  P.  C.  to  the 
several  stations  and  determine  the  station 
of  the  P.  T. 

2.  In  the  above  example  what  is  the 
tangent  distance?  the  external  distance? 
the  long  chord  ?  the  middle  ordinate  ? 


3.  In  the  above  example  what  is  the  true  length  of  the  circular  arc? 

4.  Two   tangents  intersect  at  station  182  +  75,  with  a  deflection  angle 
of  26°  30'.     It  is  required  to  connect  them  with  a  3°  30'  curve.     What  will 
be  the  station  of  the  P.  C.  and,  if  the  station  numbering  is  changed  to  read 
around  the  curve  instead  of  along  the  tangents,  what  will  be  the  station  of 
the  P.  T.  ?     (The  stationing  usually  reads  along  the  curve.) 

5.  Tabulate  the  deflections  to  each  station  on  the  curve. 

6.  It  is  required  to  connect  the  tangents  named  in  Example  4  by  a  curve 
that  shall  pass  approximately  39  feet   inside   the  vertex.      What   degree 
curve  should  be  used,  and  what  will  be  the  station  of  P.  C .  and  P.  T.  ? 


7.   A  24°  curve  is  to  be  located  by  25-foot  chords, 
deflection  for  each  chord  ? 


What  is  the  correct 


8.  If  the  24°  curve  is  to  be  so  located  that  four  chord  lengths  shall  cover 
the  arc  subtended  by  a  100-foot  chord,  what  is  the  length  of  each  chord  and 
\\hatthedeflection? 

9.  A  4°  curve  begins  at  station  376  +  50,  and  at  station  381  is  com- 
pounded to  a  6°  curve  which  ends  at  station  387  +  25.     It  is  required  to 
find  the  central  angle  of  each  arc,  the  total  central  angle,  and  the  two  tan- 
gent distances  P.  C.  to  intersection,  and  intersection  to  P.  T. 

10.  Find  all  the  offsets  necessary  to  lay  out  a  3°  curve,  with  the  chain 
alone,  by  offsets  from  the  chord  produced ;  the  curve  beginning  at  station 
367  +  30  and  ending  at  station  372  +  60. 


11.   What  is  the  degree  of  the  curve  shown  in  Fig.  156? 
Fig.  157? 


That  shown  in 


336 


APPENDIX. 


CHAPTER  IX. 

1.  By  leveling,  the  elevations  given  in  the  following  table  were  found  at 
the  points  indicated.  The  tract  is  in  the  form  of  a  square  six  hundred  feet 
on  a  side  and  divided  into  one-hundred-foot  squares.  Draw  the  map  to  a 
scale  of  100  feet  per  inch,  locating  five-foot  contours. 


STATION. 

ELEVATION. 

STATION. 

ELEVATION. 

A0 

153.0 
155.5 

B0 

154.5 
159.6 

A2 
A3 

158.0 
158.5 
156.4 

B3 

163.7 
166.5 
165.5 

A5 

152.8 

B5 

156.0 

A6 

149.5 

B6 

155.5 

co 

155.4 

D0 

155.3 

cl 

162.1 
169.0 

D0  +  80 

160.5 
160.0 

c3 

173.7 

D2 

170.2 

C3  +  50 

175.5 

D3 

180.1 

£  +  20 

170.4 
166.0 

D3  +  50 

192.2 
178.5 

C4  +  60 

167.0 

Ds 

168.7 

C5 

164.8 

D6 

161.5 

C6 

159.4 

Eo 

158.2 

F0 

157.5 

EI 

164.2 

FI 

162.3 

E2 

168.8 
175.0 

F'  +  SO 

165.5 
166.0 

E3  +  20 
Es  +  60 

176.0 
172.5 
172.5 
166.0 
161.5 

F8 

F3  +  50 
F, 
F5 
F6 

165.5 
164.0 
164.9 
163.1 
158.5 

GO 

154.5 
159.6 

G3  +  40 

155.5 
159.0 

G2 
G3 

161.0 
159.5 

G5 

157.5 
155.0 

Oc  +  50 

154.0 

3/,  +  15 

181.8 

2^  +  40 

167.5 

4B  +  75 

170.8 

3C  +  80 

182.0 

4,  +  10 

170.0 

PROBLEMS  AND  EXAMPLES. 


337 


2.   Plot  a  contour  map  with  five-foot  contours  from  the  following  notes 
of  a  stadia  survey,  all  pointings  being  taken  from  one  position. 


INST.  AT 


ON  SUMMIT  OF  BROWN'S  HILL. 


Zero  Az.  =  Mag.  North.    Elev.  Q0  assumed  100.00. 


STATION. 

AZIMUTH. 

DlST. 

VERT.  Z  DIFF.  ELEV. 

ELEV. 

1. 

357°00' 

1025 

-  2°12' 

40.0 

60.0 

2.   In    ravine    running  ) 
northerly.           ) 

355°10' 

885 

-  2°28' 

38.0 

62.0 

3. 

3°15' 

860 

-  2°20' 

34.8 

65.2 

4.   In    ravine    running  ) 
northerly.          > 

1°25' 

720 

-  2°47' 

35.0 

65.0 

5. 

15°40' 

635 

-  2°42' 

30.0 

70-0 

6. 

26°50' 

985 

-  1°58' 

33.8 

66.2 

7. 

30°00' 

1175 

-  1°48' 

37.0 

63.0 

8. 

42°00' 

1370 

-  1°42' 

40.5 

59.5 

9. 

48°30' 

1120 

-  1°45' 

34.2 

65.7 

10. 

35°20' 

730 

-  2°15' 

28.7 

71.3 

11. 

52°50' 

825 

-  2°08' 

30.7 

69.3 

12.    In    ravine    running) 
northerly.           i 

17<W 

460 

-  3°26' 

27.5 

72.5 

13. 

26°55' 

420 

-  3°25' 

25.0 

75.0 

14. 

50°05' 

470 

-  2°54' 

23.7 

76.3 

15. 

71°55' 

960 

-  2°03' 

34.2 

65.8 

16. 

72°30' 

715 

-  2°24' 

30.0 

70.0 

17. 

59°35' 

340 

-3°22 

20.0 

80.0 

18. 

30°35' 

230 

-4°47 

19.0 

81.0 

19. 

5°00' 

135 

-  6°26' 

15.0 

85.0 

20. 

48°15' 

140 

-  4°52' 

11.8 

88.2 

21. 

86°35' 

85 

-  6°48' 

10.0 

90.0 

22.   Head  of  ravine  run-  ) 
ning  easterly.       } 

94°30' 

235 

-  4°56' 

20.0 

80.0 

23. 

82°25' 

290 

-  3°45' 

19.0 

81.0 

24. 

81°15' 

545 

-  2°49' 

26.8 

73.2 

25.   In  ravine. 

95°05' 

490 

-  3°16' 

28.0 

72.0 

26.     «      " 

90°35' 

655 

-  2°50' 

32.5 

67.5 

27.     "      « 

89°05' 

835 

-  2°32' 

37.0 

63.0 

28.     "      » 

91°10' 

905 

-  2°27' 

38.8 

61.2 

29. 

102°55' 

860 

-  2°18' 

34.5 

65.5 

30. 

118°10' 

1020 

-  2°07' 

37.5 

62.5 

31. 

130°50' 

1185 

-  1°57' 

40.0 

60.0 

32. 

147°15' 

860 

-  2°12' 

33.0 

67.0 

33. 

147°15' 

580 

-  2°43' 

27.5 

72.5 

34. 

124°15' 

690 

-  2°32' 

30.5 

69.5 

35. 

143°30' 

365 

-  3°33' 

22.6 

77.4 

R'M'D  SL-RV.  —  22 

338 


APPENDIX. 


STATION. 

AZIMUTH. 

DIST. 

VERT.  £ 

DIFF.  ELEV. 

ELEV. 

36. 

118°45' 

420 

-  3°14' 

23.7 

76.3 

37. 

130°30' 

125 

-  5°33' 

12.0 

88.0 

38. 

194°00' 

140 

-7°30 

18.0 

82.0 

39. 

168°50' 

755 

-  2°28' 

32.5 

67.5 

40. 

191°55' 

615 

-  2°49' 

30.4 

69.6 

41. 

191°5.7 

315 

-  3°58' 

21.8 

78.2 

42. 

219°45' 

340 

-  4°00' 

23.6 

76.4 

43.   In   ravine    running  ) 
southwesterly.       j 

229°15' 

425 

-  4°05' 

30.0 

70.0 

44.   In    ravine    running  ) 
southwesterly.      > 

232°30' 

710 

-  3°01' 

37.5 

62.5 

45. 

223°25' 

640 

-  3°00' 

33.7 

66.3 

46. 

.   203°35' 

835 

-  2°25' 

35.2 

64.8 

47.   In  ravine. 

229°35' 

945 

-  2°31' 

41.5 

58.5 

48. 

224°15' 

1025 

-2°21' 

42.0 

58.0 

49.   In  ravine. 

229°  15' 

1185 

-  2°13' 

46.0 

54.0 

50. 

343°00' 

960 

-  2°12' 

37.0 

63.0 

51. 

331°00' 

1180 

-  1°59' 

40.5 

59.5 

52. 

320°05' 

1350 

-  1°52' 

44.0 

56.0 

53. 

315°30' 

1115 

-  2°01' 

38.9 

61.1 

54. 

326°30' 

780 

-  2°22' 

32.2 

67.8 

55. 

343°20' 

440 

-  3°16' 

25.2 

74.8 

56. 

300°05' 

795 

-  2°20' 

32.5 

67.5 

57. 

287°15' 

930 

-  1°58' 

31.8 

68.2 

58. 

2°30' 

315 

-  4°01' 

22.0 

78.0 

59. 

300°30' 

300 

-  3°55' 

20.4 

79.6 

60. 

274°45' 

770 

-  2°27' 

33.0 

67.0 

61. 

276°55' 

470 

-  2°59' 

24.5 

75.5 

62. 

340°45' 

140 

-  5°43' 

13.8 

86.2 

63. 

285°30' 

175 

-  4°37' 

14.0 

86.0 

64. 

262°55' 

275 

-  3°51' 

18.5 

81.5 

65.   Head  of  ravine 

240°20' 

225 

-  5°10' 

20.0 

80.0 

66. 

253°45' 

570 

-  2°49' 

28.0 

72.0 

67. 

243°00' 

715 

-  2°42' 

33.8 

66.2 

68. 

255°55' 

900 

-  2°25' 

38.0 

62.0 

CHAPTER   X. 

1.  Assume  in  Example  1,  Chapter  IX.,  that  the  area  shown  is  to  be  graded 
to  a  plane  surface  of  uniform  elevation  of  160  feet.     Determine  the  depth  of 
cut  or  fill  at  each  corner  of  the  small  squares  and  from  these  the  volumes. 

2.  Determine  the  volumes  by  the  approximate  graphical  method,  using 
the  planimeter  for  the  areas. 


PROBLEMS   AND   EXAMPLES. 


339 


CHAPTER   XI. 

The  following  observations1  for  rating  a  Price  acoustic  current  meter 
were  made  by  Mr.  W.  G.  Price,  June  21,  1895,  by  the  method  described  by 
him  in  "Engineering  News,"  Jan.  10,  1895.  The  student  will  use  the 
first,  fourth,  fifth,  and  sixth  columns.  The  second  and  third  columns  will 
be  understood  by  a  reference  to  the  article  in  "  Engineering  News."  Get 
the  equation  for  rating  the  meter. 

RATING  OF  PRICE  ACOUSTIC  CURRENT  METER  No.  4,  JUNE  21,  1895. 


No. 

IST  WIRE. 
Feet. 

2D  WIRE. 
Feet. 

DISTANCE. 
Feet. 

REV. 

TIME. 
Second: 

1 
2 

31.09 
24.84 

79.72 

74.28 

48.63 
48.44 

20 
20 

24.4 
22.4 

3 

37.50 

86.32 

48.82 

20 

35.0 

4 

32.76 

82.50 

49.74 

20 

56.0 

5 

26.21 

76.54 

50.33 

20 

56.0 

6 

13.61 

64.21 

50.60 

20 

99.0 

7 

27.55 

77.78 

58.23 

20 

94.6 

8 

13.45 

62.31 

48.86 

20 

25.0 

9 

23.28 

71.27 

48.09 

20 

15.8 

10 

32.22 

80.35 

48.13 

20 

10.2 

11 

27.84 

76.01 

48.17 

20 

7.4 

12 

32.08 

103.95 

71.87 

30 

20.4 

13 

39.99 

87.98 

47.99 

20 

6.0 

14 

35.08 

82.68 

47.60 

20 

5.8 

15 

24.20  . 

72.84 

48.64? 

20 

6.0 

16 

38.86 

86.39 

47.53 

20 

5.8 

17 

24.67 

72.47 

47.80 

20 

5.2 

18 

39.13 

86.85 

47.72 

20 

5.2 

19 

35.74 

83.51 

47.77  + 

20 

11.2 

20 

20.15 

67.97 

47.82  + 

20 

11.2 

21 

30.92 

79.17 

48.25 

20 

13.4 

22 

21.86 

70.30 

48.44 

20 

11.6 

23 

23.67 

72.09 

48.42  + 

20 

19.4 

24 

14.47 

62.71 

48.24  + 

20 

20.0 

25 

39.27 

88.65 

49.38 

20 

32.4 

26 

46.18 

85.82 

49.64 

20 

31.4 

27 

40.82 

90.57 

49.75 

20 

37.4 

28 

26.14 

101.15 

75.01 

30 

64.4 

The  center  of  the  meter  wheel  was  2.1  feet  below  the  water  surface.  In 
Nos.  19,  20,  23,  and  24  I  felt  sure  the  time  interval  used  to  start  the  watch 
and  stop  the  first  wire  from  passing  out,  after  I  had  heard  the  click  of  the 

i  Kindly  furnished  the  author  by  Mr.  Price. 


340  APPENDIX. 

meter,  was  longer  than  the  time  consumed  for  the  same  purpose  at  the  end 
of  the  run.  I  therefore  put  a  +  mark  to  indicate  that  the  distance  was  too 
small.  No.  15  was  doubtful,  as  the  skiff  rocked  during  the  trip.  The  six 
columns  are  an  exact  copy  of  the  field  notes. 


CHAPTER   XII. 

1.  From  the  following  data  of  the  survey  of  a  tunnel  and  shaft  for  a 
connection,  determine  the  azimuth,  length,  and  grade  of  the  connecting 
drift.     First  draw  a  map  of  the  survey  and  indicate  the  connecting  drift. 

From  a  monument  at  the  mouth  of  the  tunnel  run  in  the  tunnel,  azimuth 
36°  30',  436  feet,  vert.  Z  +  1°  00' ;  thence  azimuth  52°  10',  200  feet,  vert.  Z 
+  1°  10',  to  point  near  breast  of  tunnel. 

From  the  monument  at  the  mouth  of  the  tunnel  run  on  the  surface, 
azimuth  86°  30',  232  feet,  vert.  Z  -  3°  30' ;  thence  azimuth  40°  20',  636  feet, 
vert.  +  13°  50',  to  center  of  shaft  110  feet  deep. 

2.  The  strike  of  a  vein  of  ore  observed  on  the  point  of  an  outcrop  is 
N.  36°  W.  or  S.  36°  E.      The  dip  of  the  vein  is  found  to  be  13°  from  the 
vertical  and  to  the  northeast  (right  angle  to  the  strike).      From  a  point 
which  bears  from  the  before-mentioned  point  of  outcrop  N.  36°  E.  300  feet, 
vert.  Z  —  15°  a  cross-cut  tunnel  is  to  be  driven  level  and  in  a  direction 
S.  55°  W.     How   long  must  it  be  to  reach  the  vein  ?    What  will  be  the 
difference  if  the  tunnel  is  run  on  a  one-per-cent  grade  ? 

3.  If,  in  the  above  example,  the  tunnel  is  to  run  S.  60°  W.,  how  long 
must  it  be  ? 

4.  How  long  must  it  be,  if  the  dip  is  to  the  southwest,  the  tunnel  being 
run  S.  54°  W.  ? 

5.  How  long  must  it  be,  if  the  dip  is  to  the  southwest  and  the  tunnel  is 
run  S.  50°  W.  ? 

6.  In  Example  4  what  would  be  the  azimuth,  grade,  and  length  of  the 
shortest  tunnel  that  could  be  run  to  reach  the  vein  ? 


THE  JUDICIAL   FUNCTIONS  OF   SURVEYORS.  341 

II.    THE  JUDICIAL  FUNCTIONS  OF  SURVEYORS.1 
BY  JUSTICE  COOLEY  OF  THE  MICHIGAN  SUPREME  COURT. 

WHEN  a  man  has  had  a  training  in  one  of  the  exact  sciences,  where 
every  problem  within  its  purview  is  supposed  to  be  susceptible  of  accurate 
solution,  he  is  likely  to  be  not  a  little  impatient  when  he  is  told  that,  under 
some  circumstances,  he  must  recognize  inaccuracies,  and  govern  his  action 
by  facts  which  lead  him  away  from  the  results  which  theoretically  he  ought 
to  reach.  Observation  warrants  us  in  saying  that  this  remark  may  frequently 
be  made  of  surveyors. 

In  the  state  of  Michigan  all  our  lands  are  supposed  to  have  been  sur- 
veyed once  or  more,  and  permanent  monuments  fixed  to  determine  the 
boundaries  of  those  who  should  become  proprietors.  The  United  States,  as 
original  owner,  caused  them  all  to  be  surveyed  once  by  sworn  officers,  and 
as  the  plan  of  subdivision  was  simple,  and  was  uniform  over  a  large  extent 
of  territory,  there  should  have  been,  with  due  care,  few  or  no  mistakes ;  and 
long  rows  of  monuments  should  have  been  perfect  guides  to  the  place  of  any 
one  that  chanced  to  be  missing.  The  truth  unfortunately  is  that  the  lines 
were  very  carelessly  run,  the  monuments  inaccurately  placed ;  and,  as  the 
recorded  witnesses  to  these  were  many  times  wanting  in  permanency,  it  is 
often  the  case  that  when  the  monument  was  not  correctly  placed  it  is  im- 
possible to  determine  by  the  record,  with  the  aid  of  anything  on  the  ground, 
where  it  was  located.  The  incorrect  record  of  course  becomes  worse  than 
useless  when  the  witnesses  it  refers  to  have  disappeared. 

It  is,  perhaps,  generally  supposed  that  our  town  plots  were  more  accu- 
rately surveyed,  as  indeed  they  should  have  been,  for  in  general  there  can 
have  been  no  difficulty  in  making  them  sufficiently  perfect  for  all  practical 
purposes.  Many  of  them,  however,  were  laid  out  in  the  woods ;  some  of 
them  by  proprietors  themselves,  without  either  chain  or  compass,  and  some 
by  imperfectly  trained  surveyors,  who,  when  land  was  cheap,  did  not  appre- 
ciate the  importance  of  having  correct  lines  to  determine  boundaries  when 
land  should  become  dear.  The  fact  probably  is  that  town  surveys  are  quite 
as  inaccurate  as  those  made  under  authority  of  the  general  government. 

It  is  now  upwards  of  fifty  years  since  a  major  part  of  the  public  surveys 
in  what  is  now  the  state  of  Michigan  were  made  under  authority  of  the 
United  States.  Of  the  lands  south  of  Lansing,  it  is  now  forty  years  since 
the  major  part  were  sold  and  the  work  of  improvement  begun.  A  genera- 
tion has  passed  away  since  they  were  converted  into  cultivated  farms,  and 
few,  if  any,  of  the  original  corner  and  quarter  stakes  now  remain. 

The  corner  and  quarter  stakes  were  often  nothing  but  green  sticks  driven 
into  the  ground.  Stones  might  be  put  around  or  over  these  if  they  were 
handy,  but  often  they  were  not,  and  the  witness  trees  must  be  relied  upon 
after  the  stake  was  gone.  Too  often  the  first  settlers  were  careless  in  fixing 

1  A  paper  prepared  for  the  Michigan  Society  of  Surveyors  and  Engineers. 


342  APPENDIX. 

their  lines  with  accuracy  while  monuments  remained,  and  an  irregular 
brush  fence,  or  something  equally  untrustworthy,  may  have  been  relied 
upon  to  keep  in  mind  where  the  blazed  line  once  was.  A  fire  running 
through  this  might  sweep  it  away,  and  if  nothing  were  substituted  in  its 
place,  the  adjoining  proprietors  might  in  a  few  years  be  found  disputing 
over  their  lines,  and  perhaps  rushing  into  litigation,  as  soon  as  they  had 
occasion  to  cultivate  the  land  along  the  boundary. 

If  now  the  disputing  parties  call  in  a  surveyor,  it  is  not  likely  that  any 
one  summoned  would  doubt  or  question  that  his  duty  was  to  find,  if  pos- 
sible, the  place  of  the  original  stakes  which  determined  the  boundary  line 
between  the  proprietors.  However  erroneous  may  have  been  the  original 
survey,  the  monuments  that  were  set  must  nevertheless  govern,  even  though 
the  effect  be  to  make  one  half-quarter  section  ninety  acres  and  the  one 
adjoining  but  seventy;  for  parties  buy  or  are  supposed  to  buy  in  reference 
to  those  monuments,  and  are  entitled  to  what  is  within  their  lines,  and  no 
more,  be  it  more  or  less.  Mclver  v.  Walker,  4  Wheaton's  Reports,  444 ; 
Land  Co.  v.  Sounders,  103  U.  S.  Reports,  316;  Cottingham  v.  Parr,  93  111. 
Reports,  233 ;  Bunton  v.  Cardwell,  53  Texas  Reports,  408 ;  Watson  v.  Jones, 
85  Penn.  Reports,  117. 

While  the  witness  trees  remain  there  can  generally  be  no  difficulty  in 
determining  the  locality  of  the  stakes.  When  the  witness  trees  are  gone, 
so  that  there  is  no  longer  record  evidence  of  the  monuments,  it  is  remark- 
able how  many  there  are  who  mistake  altogether  the  duty  that  now  devolves 
upon  the  surveyor.  It  is  by  no  means  uncommon  that  we  find  men  whose 
theoretical  education  is  supposed  to  make  them  experts,  who  think  that 
when  the  monuments  are  gone  the  only  thing  to  be  done  is  to  place  new 
monuments  where  the  old  ones  should  have  been,  and  where  they  would 
have  been  if  placed  correctly.  This  is  a  serious  mistake.  The  problem  is 
now  the  same  that  it  was  before,  —  to  ascertain,  by  the  best  lights  of  which  the 
case  admits,  where  the  original  lines  were.  The  mistake  above  alluded  to  is 
supposed  to  have  found  expression  in  our  legislation ;  though  it  is  possible 
that  the  real  intent  of  the  act  to  which  we  shall  refer  is  not  what  is  com- 
monly supposed. 

An  act  passed  in  1869,  Compiled  Laws,  §  593,  amending  the  laws  respect- 
ing the  duties  and  powers  of  county  surveyors,  after  providing  for  the  case 
of  corners  which  can  be  identified  by  the  original  field  notes  or  other  un- 
questionable testimony,  directs  as  follows  : 

"  Second.  Extinct  interior  section-corners  must  be  reestablished  at  the 
intersection  of  two  right  lines  joining  the  nearest  known  points  on  the 
original  section  lines  east  and  west  and  north  and  south  of  it. 

"  Third.  Any  extinct  quarter-section  corner,  except  on  fractional  lines, 
must  be  reestablished  equidistant  and  in  a  right  line  between  the  section 
corners ;  in  all  other  cases  at  its  proportionate  distance  between  the  nearest 
original  corners  on  the  same  line." 

The  corners  thus  determined,  the  surveyors  are  required  to  perpetuate 
by  noting  bearing  trees  when  timber  is  near. 


THE  JUDICIAL   FUNCTIONS   OF   SURVEYORS.  343 

To  estimate  properly  this  legislation,  we  must  start  with  the  admitted 
and  unquestionable  fact  that  each  purchaser  from  government  bought  such 
land  as  was  within  the  original  boundaries,  and  unquestionably  owned  it  up 
to  the  time  when  the  monuments  became  extinct.  If  the  monument  was 
set  for  an  interior-section  corner,  but  did  not  happen  to  be  "  at  the  intersec- 
tion of  two  right  lines  joining  the  nearest  known  points  on  the  original 
section  lines  east  and  west  and  north  and  south  of  it,"  it  nevertheless  deter- 
mined the  extent  of  his  possessions,  and  he  gained  or  lost  according  as  the 
mistake  did  or  did  not  favor  him. 

It  will  probably  be  admitted  that  no  man  loses  title  to  his  land  or  any 
part  thereof  merely  because  the  evidences  become  lost  or  uncertain.  It  may 
become  more  difficult  for  him  to  establish  it  as  against  an  adverse  claimant, 
but  theoretically  the  right  remains ;  and  it  remains  as  a  potential  fact  so 
long  as  he  can  present  better  evidence  than  any  other  person.  And  it  may 
often  happen  that,  notwithstanding  the  loss  of  all  trace  of  a  section  corner 
or  quarter  stake,  there  will  still  be  evidence  from  which  any  surveyor  will  be 
able  to  determine  with  almost  absolute  certainty  where  the  original'boundary 
was  between  the  government  subdivisions. 

There  are  two  senses  in  which  the  word  "  extinct "  may  be  used  in  this 
connection  :  one  the  sense  of  physical  disappearance ;  the  other  the  sense  of 
loss  of  all  reliable  evidence.  If  the  statute  speaks  of  extinct  corners  in  the 
former  sense,  it  is  plain  that  a  serious  mistake  was  made  in  supposing  that 
surveyors  could  be  clothed  with  authority  to  establish  new  corners  by  an 
arbitrary  rule  in  such  cases.  As  well  might  the  statute  declare  that  if  a 
man  lose  his  deed  he  shall  lose  his  land  altogether. 

But  if  by  extinct  corner  is  meant  one  in  respect  to  the  actual  location 
of  which  all  reliable  evidence  is  lost,  then  the  following  remarks  are 
pertinent : 

(1)  There  would  undoubtedly  be  a  presumption  in  such  a  case  that  the 
corner  was  correctly  fixed  by  the  government  surveyor  where  the  field  notes 
indicated  it  to  be. 

(2)  But  this  is  only  a  presumption,  and  may  be  overcome  by  any  satis- 
factory evidence  showing  that  in  fact  it  was  placed  elsewhere. 

(3)  No  statute  can  confer  upon  a  county  surveyor  the  power  to  "estab- 
lish "  corners,  and  thereby  bind  the  parties  concerned.     Nor  is  this  a  ques- 
tion merely  of  conflict  between  state  and  federal  law ;  it  is  a  question  of 
property  right.     The   original  surveys  must  govern,  and  the   laws  under 
which  they  were  made  must  govern,  because  the  land  was  bought  in  refer- 
ence to  them ;  and  any  legislation,  whether  state  or  federal,  that  should 
have  the  effect  to  change  these,  would  be  inoperative,  because  disturbing 
vested  rights. 

(4)  In  any  case  of  disputed  lines,  unless  the  parties  concerned  settle  the 
controversy  by  agreement,  the  determination  of  it  is  necessarily  a  judicial 
act,  and  it  must  proceed  upon  evidence,  and  give  full  opportunity  for  a  hear- 
ing.    No  arbitrary  rules  of  survey  or  of  evidence  can  be  laid  down  whereby 
it  can  be  adjudged, 


344  APPENDIX. 

The  general  duty  of  a  surveyor  in  such  a  case  is  plain  enough.  He  is 
not  to  assume  that  a  monument  is  lost  until  after  he  has  thoroughly  sifted 
the  evidence  and  found  himself  unable  to  trace  it.  Even  then  he  should 
hesitate  long  before  doing  anything  to  the  disturbance  of  settled  possessions. 
Occupation,  especially  if  long  continued,  often  affords  very  satisfactory  evi- 
dence of  the  original  boundary  when  no  other  is  attainable ;  and  the  sur- 
veyor should  inquire  when  it  originated,  how,  and  why  the  lines  were  then 
located  as  they  were,  and  whether  a  claim  of  title  has  always  accompanied 
the  possession,  and  give  all  the  facts  due  force  as  evidence.  Unfortunately, 
it  is  known  that  surveyors  sometimes,  in  supposed  obedience  to  the  state 
statute,  disregard  all  evidences  of  occupation  and  claim  of  title,  and 
plunge  whole  neighborhoods  into  quarrels  and  litigation  by  assuming  to 
"  establish  "  corners  at  points  with  which  the  previous  occupation  can  not 
harmonize.  It  is  often  the  case  that  where  one  or  more  corners  are  found 
to  be  extinct,  all  parties  concerned  have  acquiesced  in  lines  which  were 
traced  by  the  guidance  of  some  other  corner  or  landmark,  which  may  or 
may  not  have  been  trustworthy ;  but  to  bring  these  lines  into  discredit  when 
the  people  concerned  do  not  question  them,  not  only  breeds  trouble  in  the 
neighborhood,  but  it  must  often  subject  the  surveyor  himself  to  annoyance 
and  perhaps  discredit,  since  in  a  legal  controversy  the  law  as  well  as  com- 
mon sense  must  declare  that  a  supposed  boundary  line  long  acquiesced  in  is 
better  evidence  of  where  the  real  line  should  be  than  any  survey  made  after 
the  original  monuments  have  disappeared.  Stewart  v.  Carleton,  31  Mich. 
Reports,  270 ;  Diehl  v.  Zanger,  39  Mich.  Reports,  601 ;  Dupont  v.  Starring, 
42  Mich.  Reports,  492.  And  county  surveyors,  no  more  than  any  others,  can 
conclude  parties  by  their  surveys. 

The  mischiefs  of  overlooking  the  facts  of  possession  must  often  appear 
in  cities  and  villages.  In  towns  the  block  and  lot  stakes  soon  disappear ; 
there  are  no  witness  trees  and  no  monuments  to  govern  except  such  as  have 
been  put  in  their  places,  or  where  their  places  were  supposed  to  be.  The 
streets  are  likely  to  be  soon  marked  off  by  fences,  and  the  lots  in  a  block 
will  be  measured  off  from  these,  without  looking  farther.  Now  it  may  per- 
haps be  known  in  a  particular  case  that  a  certain  monument  still  remaining 
was  the  starting  point  in  the  original  survey  of  the  town  plot ;  or  a  surveyor 
settling  in  the  town  may  take  some  central  point  as  the  point  of  departure 
in  his  surveys,  and  assuming  the  original  plot  to  be  accurate,  he  will  then 
undertake  to  find  all  streets  and  all  lots  by  course  and  distance  according 
to  the  plot,  measuring  and  estimating  from  his  point  of  departure.  This 
procedure  might  unsettle  every  line  and  every  monument  existing  by  acqui- 
escence in  the  town ;  it  would  be  very  likely  to  change  the  lines  of  streets, 
and  raise  controversies  everywhere.  Yet  this  is  what  is  sometimes  done ; 
the  surveyor  himself  being  the  first  person  to  raise  the  disturbing  questions. 

Suppose,  for  example,  a  particular  village  street  has  been  located  by 
acquiescence  and  use  for  many  years,  and  the  proprietors  in  a  certain  block 
have  laid  off  their  lots  in  reference  to  this  practical  location.  Two  lot 
owners  quarrel,  and  one  of  them  calls  in  a  surveyor  that  he  may  be  sure 


THE  JUDICIAL   FUNCTIONS   OF   SURVEYORS.         345 

that  his  neighbor  shall  not  get  an  inch  of  land  from  him.  This  surveyor 
undertakes  to  make  his  survey  accurate,  whether  the  original  was,  or  not, 
and  the  first  result  is,  he  notifies  the  lot  owners  that  there  is  error  in  the 
street  line,  and  that  all  fences  should  be  moved,  say,  one  foot  to  the  east. 
Perhaps  he  goes  on  to  drive  stakes  through  the  block  according  to  this  con- 
clusion. Of  course,  if  he  is  right  in  doing  this,  all  lines  in  the  village  will 
be  unsettled ;  but  we  will  limit  our  attention  to  the  single  block.  It  is  not 
likely  that  the  lot  owners  generally  will  allow  the  new  survey  to  unsettle 
their  possessions,  but  there  is  always  a  probability  of  finding  some  one  dis- 
posed to  do  so.  We  shall  then  have  a  lawsuit;  and  with  what  result? 

It  is  a  common  error  that  lines  do  not  become  fixed  by  acquiescence 
in  a  less  time  than  twenty  years.  In  fact,  by  statute,  road  lines  may  be- 
come conclusively  fixed  in  ten  years ;  and  there  is  no  particular  time  that 
shall  be  required  to  conclude  private  owners,  where  it  appears  that  they 
have  accepted  a  particular  line  as  their  boundary,  and  all  concerned  have 
cultivated  and  claimed  up  to  it.  McNamara  v.  Seaton,  82  111.  Reports,  498; 
Bunce  v.  Bidwell,  43  Mich.  Reports,  542.  Public  policy  requires  that  such 
lines  be  not  lightly  disturbed,  or  disturbed  at  all  after  the  lapse  of  any  con- 
siderable time.  The  litigant,  therefore,  who  in  such  a  case  pins  his  faith  on 
the  surveyor,  is  likely  to  suffer  for  his  reliance,  and  the  surveyor  himself  to 
be  mortified  by  a  result  that  seems  to  impeach  his  judgment. 

Of  course  nothing  in  what  has  been  said  can  require  a  surveyor  to 
conceal  his  own  judgment,  or  to  report  the  facts  one  way  when  he  believes 
them  to  be  another.  He  has  no  right  to  mislead,  and  he  may  rightfully 
express  his  opinion  that  an  original  monument  was  at  one  place,  when  at 
the  same  time  he  is  satisfied  that  acquiescence  has  fixed  the  rights  of  parties 
as  if  it  were  at  another.  But  he  would  do  mischief  if  he  were  to  attempt 
to  "establish"  monuments  which  he  knew  would  tend  to  disturb  settled 
rights ;  the  farthest  he  has  a  right  to  go,  as  an  officer  of  the  law,  is  to  ex- 
press his  opinion  where  the  monument  should  be,  at  the  same  time  that  he 
imparts  the  information  to  those  who  employ  him,  and  who  might  otherwise 
be  misled,  that  the  same  authority  that  makes  him  an  officer  and  entrusts 
him  to  make  surveys,  also  allows  parties  to  settle  their  own  boundary  lines, 
and  considers  acquiescence  in  a  particular  line  or  monument,  for  any  con- 
siderable period,  as  strong,  if  not  conclusive,  evidence  of  such  settlement. 
The  peace  of  the  community  absolutely  requires  this  rule.  Joyce  v.  Williams, 
26  Mich.  Reports,  332.  It  is  not  long  since  that,  in  one  of  the  leading  cities 
of  the  state,  an  attempt  was  made  to  move  houses  two  or  three  rods  into 
a  street,  on  the  ground  that  a  survey  under  which  the  street  had  been  lo- 
cated for  many  years  had  been  found  on  more  recent  survey  to  be  erroneous. 

From  the  foregoing  it  will  appear  that  the  duty  of  the  surveyor  where 
boundaries  are  in  dispute  must  be  varied  by  the  circumstances.  (1)  He 
is  to  search  for  original  monuments,  or  for  the  places  where  they  were 
originally  located,  and  allow  these  to  control  if  he  finds  them,  unless  he  has 
reason  to  believe  that  agreements  of  the  parties,  express  or  implied,  have 
rendered  them  unimportant.  By  monuments  in  the  case  of  government  sur- 


346  APPENDIX. 

veys,  we  mean,  of  course,  the  corner  and  quarter  stakes :  blazed  lines  or 
marked  trees  on  the  lines  are  not  monuments;  they  are  merely  guides  or 
finger  posts,  if  we  may  use  the  expression,  to  inform  us  with  more  or  less 
accuracy  where  the  monuments  may  be  found.  (2)  If  the  original  mon- 
uments are  no  longer  discoverable,  the  question  of  location  becomes  one  of 
evidence  merely.  It  is  merely  idle  for  any  state  statute  to  direct  a  surveyor 
to  locate  or  "establish"  a  corner,  as  the  place  of  the  original  monument, 
according  to  some  inflexible  rule.  The  surveyor,  on  the  other  hand,  must 
inquire  into  all  the  facts;  giving  due  prominence  to  the  acts  of  parties  con- 
cerned, and  always  keeping  in  mind,jirst,  that  neither  his  opinion  nor  his 
survey  can  be  conclusive  upon  parties  concerned ;  second,  that  courts  and 
juries  may  be  required  to  follow  after  the  surveyor  over  the  same  ground, 
and  that  it  is  exceedingly  desirable  that  he  govern  his  action  by  the  same 
lights  and  rules  that  will  govern  theirs.  On  town  plots  if  a  surplus  or  defi- 
ciency appears  in  a  block,  when  the  actual  boundaries  are  compared  with 
the  original  figures,  and  there  is  no  evidence  to  fix  the  exact  location  of 
the  stakes  which  marked  the  division  into  lots,  the  rule  of  common  sense 
and  of  law  is  that  the  surplus  or  deficiency  is  to  be  apportioned  between  the 
lots,  on  an  assumption  that  the  error  extended  alike  to  all  parts  of  the 
block.  O'Brien  v.  McGrane,  29  Wis.  Reports,  446  ;  Quinnin  v.  Reixers,  46 
Mich.  Reports,  605. 

It  is  always  possible  when  corners  are  extinct  that  the  surveyor  may 
usefully  act  as  a  mediator  between  parties,  and  assist  in  preventing  legal 
controversies  by  settling  doubtful  lines.  Unless  he  is  made  for  this  purpose 
an  arbitrator  by  legal  submission,  the  parties,  of  course,  even  if  they  consent 
to  follow  his  judgment,  can  not,  on  the  basis  of  mere  consent,  be  compelled 
to  do  so;  but  if  he  brings  about  an  agreement,  and  they  carry  it  into  effect 
by  actually  conforming  their  occupation  to  his  lines,  the  action  will  conclude 
them.  Of  course  it  is  desirable  that  all  such  agreements  be  reduced  to 
writing;  but  this  is  not  absolutely  indispensable  if  they  are  carried  into 
effect  without. 

Meander  Lines.  —  The  subject  to  which  allusion  will  now  be  made  is 
taken  up  with  some  reluctance,  because  it  is  believed  the  general  rules  are 
familiar.  Nevertheless  it  is  often  found  that  surveyors  misapprehend  them, 
or  err  in  their  application;  and  as  other  interesting  topics  are  somewhat 
connected  with  this,  a  little  time  devoted  to  it  will  probably  not  be  altogether 
lost.  The  subject  is  that  of  meander  lines.  These  are  lines  traced  along 
the  shores  of  lakes,  ponds,  and  considerable  rivers  as  the  measures  of 
quantity  when  sections  are  made  fractional  by  such  waters.  These  have 
determined  the  price  to  be  paid  when  government  lands  were  bought,  and 
perhaps  the  impression  still  lingers  in  some  minds  that  the  meander  lines 
are  boundary  lines,  and  all  in  front  of  them  remains  unsold.  Of  course  this 
is  erroneous.  There  was  never  any  doubt  that,  except  on  the  large  navi- 
gable rivers,  the  boundary  of  the  owners  of  the  banks  is  the  middle  line  of 
the  river;  and  while  some  courts  have  held  that  this  was  the  rule  on  all 
fresh-water  streams,  large  and  small,  others  have  held  to  the  doctrine  that 


THE   JUDICIAL   FUNCTIONS   OF   SURVEYORS.  347 

the  title  to  the  bed  of  the  stream  below  low-water  mark  is  in  the  state, 
while  conceding  to  the  owners  of  the  banks  all  riparian  rights.  The  prac- 
tical difference  is  not  very  important.  In  this  state  the  rule  that  the  center 
line  is  the  boundary  line  is  applied  to  all  our  great  rivers,  including  the 
Detroit,  varied  somewhat  by  the  circumstance  of  there  being  a  distinct 
channel  for  navigation  in  some  cases  with  the  stream  in  the  main  shallow, 
and  also  sometimes  by  the  existence  of  islands. 

The  troublesome  questions  for  surveyors  present  themselves  when  the 
boundary  line  between  two  contiguous  estates  is  to  be  continued  from  the 
meander  line  to  the  center  line  of  the  river.  Of  course  the  original  survey 
supposes  that  each  purchaser  of  land  on  the  stream  has  a  water  front  of  the 
length  shown  by  the  field  notes ;  and  it  is  presumable  that  he  bought  this 
particular  land  because  of  that  fact.  In  many  cases  it  now  happens  that 
the  meander  line  is  left  some  distance  from  the  shore  by  the  gradual  change 
of  course  of  the  stream  or  diminution  of  the  flow  of  water.  Now  the  divid- 
ing line  between  two  government  subdivisions  might  strike  the  meander 
line  at  right  angles,  or  obliquely ;  and  in  some  cases,  if  it  were  continued  in 
the  same  direction  to  the  center  line  of  the  river,  might  cut  off  from  the 
water  one  of  the  subdivisions  entirely,  or  at  least  cut  it  off  from  any  privi- 
lege of  navigation,  or  other  valuable  use  of  the  water,  while  the  other  might 
have  a  water  front  much  greater  than  the  length  of  a  line  crossing  it  at 
right  angles  to  its  side  lines.  The  effect  might  be  that,  of  two  government 
subdivisions  of  equal  size  and  cost,  one  would  be  of  very  great  value  as 
water  front  property,  and  the  other  comparatively  valueless.  A  rule  which 
would  produce  this  result  would  not  be  just,  and  it  has  not  been  recognized 
in  the  law. 

Nevertheless  it  is  not  easy  to  determine  what  ought  to  be  the  correct 
rule  for  every  case.  If  the  river  has  a  straight  course,  or  one  nearly  so, 
every  man's  equities  will  be  preserved  by  this  rule :  Extend  the  line  of  divi- 
sion between  the  two  parcels  from  the  meander  line  to  the  center  line  of 
the  river,  as  nearly  as  possible  at  right  angles  to  the  general  course  of  the 
river  at  that  point.  This  will  preserve  to  each  man  the  water  front  which 
the  field  notes  indicated,  except  as  changes  in  the  water  may  have  affected 
it,  and  the  only  inconvenience  will  be  that  the  division  line  between  differ- 
ent subdivisions  is  likely  to  be  more  or  less  deflected  where  it  strikes  the 
meander  line. 

This  is  the  legal  rule,  and  it  is  not  limited  to  government  surveys,  but 
applies  as  well  to  water  lots  which  appear  as  such  on  town  plots.  Bay  City 
Gas  Light  Co.  v.  The  Industrial  Works,  28  Mich.  Reports,  182.  It  often 
happens,  therefore,  that  the  lines  of  city  lots  bounded  on  navigable  streams 
are  deflected  as  they  strike  the  bank,  or  the  line  where  the  bank  was  when 
the  town  was  first  laid  out. 

When  the  stream  is  very  crooked,  and  especially  if  there  are  short 
bends,  so  that  the  foregoing  rule  is  incapable  of  strict  application,  it  is 
sometimes  very  difficult  to  determine  what  shall  be  done;  and  in  many 
cases  the  surveyor  may  be  under  the  necessity  of  working  out  a  rule  for 


348  APPENDIX. 

himself.  Of  course  his  action  can  not  be  conclusive;  but  if  he  adopts  one 
that  follows,  as  nearly  as  the  circumstances  will  admit,  the  general  rule 
above  indicated,  so  as  to  divide,  as  near  as  may  be,  the  bed  of  the  stream 
among  the  adjoining  owners  in  proportion  to  their  lines  upon  the  shore,  his 
division,  being  that  of  an  expert,  made  upon  the  ground  and  with  all  avail- 
able lights,  is  likely  to  be  adopted  as  law  for  the  case.  Judicial  decisions, 
into  which  the  surveyor  would  find  it  prudent  to  look  under  such  circum- 
stances, will  throw  light  upon  his  duties  and  may  constitute  a  sufficient  guide 
when  peculiar  cases  arise.  Each  riparian  lot  owner  ought  to  have  a  line  on 
the  legal  boundary,  namely,  the  center  line  of  the  stream,  proportioned  to 
the  length  of  his  line  on  the  shore ;  and  the  problem  in  each  case  is,  how 
this  is  to  be  given  him.  Alluvion,  when  a  river  imperceptibly  changes  its 
course,  will  be  apportioned  by  the  same  rules. 

The  existence  of  islands  in  a  stream,  when  the  middle  line  constitutes 
a  boundary,  will  not  affect  the  apportionment  unless  the  islands  were  sur- 
veyed out  as  government  subdivisions  in  the  original  admeasurement. 
Wherever  that  was  the  case,  the  purchaser  of  the  island  divides  the  bed  of 
the  stream  on  each  side  with  the  owner  of  the  bank,  and  his  rights  also 
extend  above  and  below  the  solid  ground,  and  are  limited  by  the  peculiari- 
ties of  the  bed  and  the  channel.  If  an  island  was  not  surveyed  as  a  govern- 
ment subdivision  previous  to  the  sale  of  the  bank,  it  is  of  course  impossible 
to  do  this  for  the  purposes  of  governmental  sale  afterwards,  for  the  reason 
that  the  rights  of  the  bank  owners  are  fixed  by  their  purchase  :  when  mak- 
ing that,  they  have  a  right  to  understand  that  all  land  between  the  meander 
lines,  not  separately  surveyed  and  sold,  will  pass  with  the  shore  in  the  gov- 
ernment sale ;  and  having  this  right,  anything  which  their  purchase  would 
include  under  it  can  not  afterward  be  taken  from  them.  It  is  believed,  how- 
ever, that  the  federal  courts  would  not  recognize  the  applicability  of  this 
rule  to  large  navigable  rivers,  such  as  those  uniting  the  Great  Lakes. 

On  all  the  little  lakes  of  the  state  which  are  mere  expansions  near  their 
mouths  of  the  rivers  passing  through  them  —  such  as  the  Muskegon,  Pere 
Marquette,  and  Manistee  —  the  same  rule  of  bed  ownership  has  been  judi- 
cially applied  that  is  applied  to  the  rivers  themselves ;  and  the  division  lines 
are  extended  under  the  water  in  the  same  way.  Rice  v.  Ruddiman,  10 
Mich.  125.  If  such  a  lake  were  circular,  the  lines  would  converge  to  the 
center ;  if  oblong  or  irregular,  there  might  be  a  line  in  the  middle  on  which 
they  would  terminate,  whose  course  would  bear  some  relation  to  that  of  the 
shore.  But  it  can  seldom  be  important  to  follow  the  division  line  very  far 
under  the  water,  since  all  private  rights  are  subject  to  the  public  rights  of 
navigation  and  other  use,  and  any  private  use  of  the  lands  inconsistent  with 
these  would  be  a  nuisance,  and  punishable  as  such.  It  is  sometimes  im- 
portant, however,  to  run  the  lines  out  for  some  considerable  distance,  in 
order  to  determine  where  one  may  lawfully  moor  vessels  or  rafts,  for  the 
winter,  or  cut  ice.  The  ice  crop  that  forms  over  a  man's  land  of  course 
belongs  to  him.  Lorman  v.  Benson,  8  Mich.  18 ;  People's  Ice  Co.  v.  Steamer 
Excelsior,  recently  decided. 


THE   JUDICIAL   FUNCTIONS   OF   SURVEYORS.  349 

What  is  said  above  will  show  how  unfounded  is  the  notion,  which  is 
sometimes  advanced,  that  a  riparian  proprietor  on  a  meandered  river  may 
lawfully  raise  the  water  in  the  stream  without  liability  to  the  proprietors 
above,  provided  he  does  not  raise  it  so  that  it  overflows  the  meander  line. 
The  real  fact  is  that  the  meander  line  has  nothing  to  do  with  such  a  case, 
and  an  action  will  lie  whenever  he  sets  back  the  water  upon  the  proprietor 
above,  whether  the  overflow  be  below  the  meander  lines  or  above  them. 

As  regards  the  lakes  and  ponds  of  the  state,  one  may  easily  raise  ques- 
tions that  it  would  be  impossible  for  him  to  settle.  Let  us  suggest  a  few 
questions,  some  of  which  are  easily  answered,  and  some  not: 

(1)  To  whom  belongs  the  land  under  these  bodies  of  water,  where  they 
are  not  mere  expansions  of  a  stream  flowing  through  them  ? 

(2)  What  public  rights  exist  in  them? 

(3)  If  there  are  islands  in  them  which  were  not  surveyed  out  and  sold  by 
the  United  States,  can  this  be  -done  now  ? 

Others  will  be  suggested  by  the  answers  given  to  these. 

It  seems  obvious  that  the  rules  of  private  ownership  which  are  applied  to 
rivers  can  not  be  applied  to  the  Great  Lakes.  Perhaps  it  should  be  held  that 
the  boundary  is  at  low-water  mark,  but  improvements  beyond  this  would 
only  become  unlawful  when  they  became  nuisances.  Islands  in  the  Great 
Lakes  would  belong  to  the  United  States  until  sold,  and  might  be  surveyed 
and  measured  for  sale  at  any  time.  The  right  to  take  fish  in  the  lakes,  or 
to  cut  ice,  is  public  like  the  right  of  navigation,  but  is  to  be  exercised  in  such 
manner  as  not  to  interfere  with  the  rights  of  shore  owners.  But  so  far  as 
these  public  rights  can  be  the  subject  of  ownership,  they  belong  to  the  state, 
not  to  the  United  States ;  and,  so  it  is  believed,  does  the  bed  of  a  lake  also. 
Pollard  v.  Hagan,  3  Howard's  U.  S.  Reports.  But  such  rights  are  not  gener- 
ally considered  proper  subjects  of  sale,  but,  like  the  right  to  make  use  of  the 
public  highways,  they  are  held  by  the  state  in  trust  for  all  the  people. 

What  is  said  of  the  large  lakes  may  perhaps  be  said  also  of  many  of  the 
interior  lakes  of  the  state ;  such,  for  example,  as  Houghton,  Higgins,  Che- 
boygan,  Burt's,  Mullet,  Whitmore,  and  many  others.  But  there  are  many 
little  lakes  or  ponds  which  are  gradually  disappearing,  and  the  shore  pro- 
prietorship advances  part  passu  as  the  waters  recede.  If  these  are  of  any 
considerable  size  —  say,  even  a  mile  across  — there  may  be  questions  of  con- 
flicting rights  which  no  adjudication  hitherto  made  could  settle.  Let  any 
surveyor,  for  example,  take  the  case  of  a  pond  of  irregular  form,  occupying 
a  mile  square  or  more  of  territory,  and  undertake  to  determine  the  rights 
of  the  shore  proprietors  to  its  bed  when  it  shall  totally  disappear,  and  he 
will  find  he  is  in  the  midst  of  problems  such  as  probably  he  has  never 
grappled  with,  or  reflected  upon  before.  But  the  general  rules  for  the  exten- 
sion of  shore  lines,  which  have  already  been  laid  down,  should  govern  such 
cases,  or  at  least  should  serve  as  guides  in  their  settlement.1 

1  Since  this  address  was  delivered,  some  of  these  questions  have  received  the 
attention  of  the  Supreme  Court  of  Michigan  in  the  cases  of  Richardson  v.  Prentiss, 
48  Mich.  Reports,  88,  and  Backus  v.  Detroit,  Albany  Law  Journal,  vol.  26,  p.  428. 


350  APPENDIX. 

Where  a  pond  is  so  small  as  to  be  included  within  the  lines  of  a  private 
purchase  from  the  government,  it  is  not  believed  the  public  have  any  rights 
in  it  whatever.  Where  it  is  not  so  included,  it  is  believed  they  have  rights 
of  fishery,  rights  to  take  ice  and  water,  and  rights  of  navigation  for  business 
or  pleasure.  This  is  the  common  belief,  and  probably  the  just  one.  Shore 
rights  must  not  be  so  exercised  as  to  disturb  these,  and  the  states  may  pass 
all  proper  laws  for  their  protection .  It  would  be  easy  with  suitable  legislation 
to  preserve  these  little  bodies  of  water  as  permanent  places  of  resort  for  the 
pleasure  and  recreation  of  the  people,  and  there  ought  to  be  such  legislation. 

If  the  state  should  be  recognized  as  owner  of  the  beds  of  these  small 
lakes  and  ponds,  it  would  not  be  owner  for  the  purpose  of  selling.  It  would 
be  owner  only  as  a  trustee  for  the  public  use ;  and  a  sale  would  be  incon- 
sistent with  the  right  of  the  bank  owners  to  make  use  of  the  water  in  its 
natural  condition  in  connection  writh  their  estates.  Some  of  them  might  be 
made  salable  lands  by  draining ;  but  the  state  could  not  drain,  even  for  this 
purpose,  against  the  will  of  the  shore  owners,  unless  their  rights  were  appro- 
priated and  paid  for. 

Upon  many  questions  that  might  arise  between  the  state  as  owner  of  the 
bed  of  a  little  lake  and  the  shore  owners,  it  would  be  presumptuous  to 
express  an  opinion  now,  and  fortunately  the  occasion  does  not  require  it. 

I  have  thus  indicated  a  few  of  the  questions  with  which  surveyors  may 
now  and  then  have  occasion  to  deal,  and  to  which  they  should  bring  good 
sense  and  sound  judgment.  Surveyors  are  not  and  can  not  be  judicial  offi- 
cers, but  in  a  great  many  cases  they  act  in  a  quasi  judicial  capacity  with  the 
acquiescence  of  parties  concerned ;  and  it  is  important  for  them  to  know  by 
what  rules  they  are  to  be  guided  in  the  discharge  of  their  judicial  functions. 
What  I  have  said  can  not  contribute  much  to  their  enlightenment,  but  I 
trust  will  not  be  wholly  without  value. 


THE   OWNERSHIP   OF   SURVEYS,   ETC.  351 


III.  THE  OWNERSHIP  OF  SURVEYS,  AND  WHAT  CONSTITUTES 
A  SURVEY  AND  MAP.1 

THERE  seems  to  be  a  difference  of  opinion  among  surveyors  as  to  how 
much  of  the  information  obtained,  and  how  much  of  the  work  done  in  mak- 
ing a  survey  shall  be  furnished  to  the  individual  for  whom  the  survey  is 
made.  Many  surveyors  keep  what  are  called  "private  notes."  All  men 
doing  business  as  surveyors  keep  notes  of  all  surveys  in  a  convenient  form 
for  ready  reference.  The  extent  to  which  these  notes  are  private  has  not 
been  rightly  comprehended  by  all  surveyors,  and  hence  has  resulted  the 
difference  of  opinion  mentioned. 

This  article  is  an  attempt  to  present  a  side  of  this  question  that  has  not 
heretofore  been  fully  considered.  An  endeavor  has  also  been  made  to  point 
out  to  the  young  surveyor  a  line  of  action  expedient  for  him  to  follow,  which 
will  at  the  same  time  be  found  advantageous  to  the  community  in  which  he 
works. 

In  this  discussion  the  question  arises  at  once,  "What  constitutes  a  sur- 
vey?" and  the  answer  obviously  depends  on  the  object  of  the  survey.  This 
discussion  will  be  confined  to  land  surveys ;  that  is,  to  surveys  made  for  the 
purpose  (1)  of  subdividing  a  large  tract  of  land  into  smaller  parcels  to  be 
sold  ;  (2)  of  determining  the  boundary  of  a  tract  the  description  of  which  is 
known ;  (3)  of  determining  the  description  when  the  boundaries  are  known. 

The  principle  to  be  enunciated  applies  to  any  other  survey  as  well,  be  it 
railroad,  canal,  bridge,  or  topographical  survey.  Indeed,  it  is  well  under- 
stood in  all  such  surveys,  but  seems  to  be  ignored  by  many  engineers  having 
to  do  with  land  surveys. 

A  survey  is  the  operation  of  finding  the  contour,  dimensions,  position, 
or  other  particulars  of  any  part  of  the  earth's  surface,  .  .  .  tract  of  land, 
etc.,  and  representing  the  same  on  paper. 

In  making  a  survey  it  is  necessary  to  set  certain  points,  called  monu- 
ments or  corners,  and  to  determine  a  description  of  these  points.  These 
items  therefore  become  a  part  of  the  survey.  Then  a  map  must  be  drawn. 
This  map,  to  be  a  faithful  representation  of  the  ground  and  the  work  done, 
should,  together  with  the  notes,  show  all  of  the  items  mentioned. 

The  object  of  establishing  monuments  or  corners  and  describing  them 
is  twofold :  (1)  to  mark  on  the  ground  the  boundaries  of  the  tract,  and 
(2)  to  secure  definite  information  as  to  its  location  with  reference  to  other 
points  or  tracts,  so  that  from  this  information  the  land  may  at  a  future 
time  be  found.  For  a  complete  survey,  therefore,  the  corners  must  be  fixed, 
information  that  will  preserve  their  location  must  be  obtained,  and  the  facts 
must  be  delineated  on  a  map  with  accompanying  notes. 

To  whom  belongs  this  survey?  It  would  appear  to  be  evident  that  it 
belongs  to  the  individual  who  pays  to  have  it  made.  It  is  not  readily  seen  in 

1  A  paper  prepared  by  the  author  for  "The  Polytechnic,"  the  student  journal  of 
the  Rensselaer  Polytechnic  Institute,  from  which  it  is  taken. 


352  APPENDIX. 

what  way  the  survey,  or  any  part  of  it,  becomes  the  sole  property  of  the  sur- 
veyor. He  may  keep  a  copy  of  his  notes  to  facilitate  his  future  work ;  but  he 
has  not  the  shadow  of  a  claim  to  a  single  note,  the  time  for  taking  which 
has  been  paid  for  by  his  employer.  If  his  charge  for  his  work  is  on  a  time 
basis,  there  can  be  no  question  as  to  the  correctness  of  the  above  statements. 
If  he  contracts  to  do  the  work  for  a  definite  sum  for  the  entire  job,  he  may 
take  as  much  time  as  he  likes,  and  may  keep  as  many  private  notes  as  he 
desires ;  but  he  is  bound  in  honor  to  return  to  his  employer  the  complete 
survey;  and,  if  he  does  so,  the  private  notes  would  thereafter  be  of  no 
great  assistance  to  him  in  securing  further  employment,  particularly  when 
it  is  remembered  that  men  of  repute  do  not  bid  against  each  other  for 
professional  work.  His  reputation  for  accuracy  and  honesty  will  be  a  far 
more  potent  factor  in  securing  employment  than  any  set  of  private  notes 
fairly  obtained. 

It  is  true  that  a  great  many  surveyors  hold  a  different  opinion,  and 
purposely  return  their  maps  and  notes  in  such  condition,  that,  while  they 
may  serve  the  purpose  for  which  they  are  primarily  made,  they  do  not  tell 
the  whole  story,  nor  enough  to  make  it  easy  for  another  surveyor  to  relocate 
the  tract  surveyed.  When  this  is  done,  the  person  ordering  the  survey  does 
not  receive  what  he  pays  for.  Something  is  withheld.  It  seems  to  need  no 
argument  to  show  that  this  is  radically  wrong. 

But  there  is  another  reason  for  condemning  this  practice.  The  correct 
and  permanent  location  of  all  public  land  lines,  as  streets,  alleys,  etc.,  as 
well  as  the  permanent  location  of  party  lines  between  private  owners,  is  a 
matter  of  the  gravest  importance,  and  no  information  that  will  at  all  serve 
to  fix  such  lines  in  their  correct  positions  for  all  time,  should  be  withheld 
from  the  owner  who  pays  for  the  survey,  be  it  private  citizen,  municipality, 
county,  or  state. 

The  records  of  monuments  and  street  lines  made  by  a  city  engineer  are 
no  more  his  private  property  than  are  the  records  in  the  offices  of  the  clerk, 
auditor,  or  treasurer,  the  property  of  the  individuals  who  held  office  at  the 
time  the  records  were  made.  The  correctness  of  the  position  assumed  has 
been  indicated  by  court  decisions. 

A  great  deal  of  laxity  is  shown  in  the  conduct  of  offices  of  city  engineers 
and  county  surveyors.  The  methods  of  regulating  the  pay  of  these  officers 
has  doubtless  had  much  to  do  with  this.  It  is  frequently  the  case  that 
the  surveyor  receives  no  salary,  but  is  allowed  to  collect  certain  specified 
fees  for  work  performed,  and  this  gives  color  to  his  claim  that  his  work 
is  private  and  belongs  to  him.  That  this  is  not  true  concerning  the  public 
work  he  does  is  evident  from  what  has  preceded.  That  the  records  of  work 
done  for  private  citizens  are  not  the  property  of  the  public  needs  no  demon- 
stration ;  but  such  work  belongs  to  those  citizens  for  whom  it  was  done. 

A  different  policy  should  be  pursued  with  regard  to  these  offices.  In 
every  case  such  an  office  should  be  a  salaried  one,  with  such  salaried  assistants 
as  may  be  necessary.  Certain  fees  should  be  prescribed  for  performing  the 
various  kinds  of  work  that  the  surveyor  may  be  called  upon  to  do  within  the 


THE   OWNERSHIP  OF   SURVEYS,   ETC.  353 

limits  of  the  territory  of  the  political  division  whose  servant  he  is.  These 
fees  should  cover  all  work  connected  with  public  construction  and  public  or 
private  land  lines,  and  should  be  returned  to  the  public  treasury. 

Their  amount  may  be  regulated,  from  time  to  time,  so  that  they  shall 
aggregate  a  sum  sufficient  to  pay  the  expenses  of  the  office.  They  should,  of 
course,  not  cover  work  of  a  private  character  not  having  to  do  with  land 
lines.  But  the  entire  public  is  interested  in  the  permanency  of  land  lines, 
and  all  records  concerning  them  made  by  a  public  official  should  become 
public  property.  The  permanency  of  land  lines  is  too  important  a  matter 
to  be  subject  to  avaricious  and  jealous  rivalry ;  and  all  the  surveyors  in  a 
given  district  should  cooperate  to  preserve,  in  their  correct  places,  all  lines 
within  the  district. 

To  this  end,  the  returns  of  every  surveyor  made  to  the  owner  should  be 
thoroughly  complete.  Maps  made  for  filing  as  public  records  should  be  so 
finished  as  to  enable  any  surveyor  to  relocate  the  land  without  the  least 
uncertainty  as  to  the  correctness  of  his  work.  That  this  is  done  in  very 
few  instances  is  well  known  to  every  surveyor  who  has  had  occasion  to 
examine  public  records  for  data  for  surveys  which  he  has  been  called  upon 
to  make. 

Because  of  the  fact  that  in  most  cases  neither  owners  nor  attorneys 
have  been  fully  posted  as  to  what  constitutes  a  complete  description,  suffi- 
cient for  relocation,  and  because  surveyors  have  been  willing  to  let  matters 
stand  as  they  were,  great  carelessness  has  arisen  in  the  practice  of  making 
and  filing  maps  for  record. 

While  in  some  states  good  laws  exist  prescribing  what  shall  appear  on 
a  map  before  it  will  be  received  as  a  public  record,  in  more  states  there  is 
nothing  whatever  to  guide  either  owner,  surveyor,  attorney,  or  recorder 
in  the  matter.  In  the  county  records  in  such  states,  anything  that  is  made 
up  of  lines  and  figures  and  labeled  "  this  is  a  map,"  is  considered  a  sufficient 
basis  for  a  correct  description  and  location  of  the  property  it  purports  to 
represent  —  no  matter  whether  it  is  drawn  by  hand,  photo-lithographed,  or 
simply  printed  with  "  rule  "  and  type.  The  records  are  full  of  auctioneers' 
circulars,  manufactured  in  a  printing  office  from  information  coming  from 
nobody  knows  where,  filed  at  the  request  of  the  auctioneer's  clerk,  with  no 
name  of  owner  or  other  interested  party  attached,  except  as  the  name  of  the 
auctioneer  appears  in  the  accompanying  advertisement.  Further  than  this, 
these  maps  are  frequently  purposely  distorted  to  create  a  favorable  impres- 
sion of  the  property  to  be  sold.  Wide  streets  are  shown  where  only  narrow 
ones  exist,  streets  appear  opened  for  the  full  width  where  they  have  been 
opened  for  but  half  their  width,  subdivisions  are  indicated  as  rectangles 
that  really  may  not  be  even  parallelograms,  etc.  Such  maps  as  these 
frequently  form  the  only  basis  for  the  description  and  location  of  the  prop- 
erty they  are  supposed  to  represent.  Such  misrepresentations  are  bad,  very 
bad  for  those  who  buy;  but  is  the  information  given  by  these  circulars 
much  worse  than  that  furnished  by  many  of  the  maps  made  by  surveyoi'S 
and  filed  at  the  request  of  the  owners? 
K'M'D  SURV.  — 23 


354  APPENDIX. 

On  these  plots,  if  of  "  additions,"  we  find  lines  indicating  the  boundaries 
of  blocks  and  lots,  all  of  which  blocks  and  lots  are  numbered;  the  names  of 
streets  appear  in  neat  letters;  a  few  dimensions,  possibly  all  linear  dimen- 
sions, will  be  given;  the  streets  or  blocks  may  be  delicately  tinted,  and 
the  whole  set  off  with  a  fine  border  and  title.  As  an  exhibition  of  the 
draughtsman's  skill,  these  maps  are  perhaps  valuable.  As  a  source  of  in- 
formation as  to  the  location  of  the  lines  they  purport  to  show,  they  are 
worth  little  more  than  the  auctioneer's  circular.  Perhaps  they  have  a  few 
more  figures,  and  the  presumption  may  be  a  little  stronger  that  the  figures 
are  correct. 

Examine  one  of  these  maps  closely.  There  will  be  found  no  evidence 
that  a  monument  has  been  set  in  the  field;  not  an  angle  is  recorded, 
though  the  lines  may  cross  at  all  sorts  of  angles ;  and  dimensions  are  given 
that  do  not  agree  among  themselves,  so  that  the  angles  can  not  be  cal- 
culated. There  will  be  found  no  name  signed  except,  possibly,  that  of  the 
surveyor,  who  thus  advertises  what  we  shall  charitably  call  his  stupidity. 

Frequently  no  monuments  are  set  except  small  stakes  at  the  corners 
of  the  blocks;  but  even  the  fact  that  such  stakes  have  been  set  is  not 
recorded  on  the  plot.  One  who  is  acquainted  with  the  practice  of  surveyors 
in  a  given  district  knows  at  what  points  to  look  for  such  stakes,  and  if  they 
have  been  set  and  not  pulled  out  to  make  room  for  a  fence  post  or  building, 
he  may  succeed  in  finding  them.  Some  surveyors  are  accustomed  to  set 
stakes  a  certain  distance  away  from  the  point  the  stake  is  supposed  to  mark, 
but  no  mention  of  this  fact  appears  on  the  map.  In  fact,  the  map  is  so 
drawn  that  no  one  but  the  surveyor  who  made  it  can  write  a  description  of 
any  one  of  the  parcels  of  land  shown,  or  correctly  locate  it  on  the  ground. 
Furthermore,  the  surveyor  himself  finds  it  impossible,  after  the  lapse  of  a 
few  years  and  the  destruction  of  his  "private  marks,"  to  rerun  any  one  of 
the  lines  exactly  as  originally  laid  out. 

It  is  easy  to  see  to  what  this  leads  —  impossible  descriptions  of  property, 
giving  opportunity  for  differences  in  judgment  as  to  interpretation  of  what 
was  intended;  disputes  as  to  position  of  party  lines;  costly  litigation  and 
expensive  movement  of  structures  begun  or  completed ;  and  the  actual 
shifting  of  lines  back  and  forth  by  different  surveyors,  or  even  by  the  same 
surveyor,  honestly  trying  to  locate  the  lines  properly. 

The  writer  has  seen  enough  trouble  of  this  sort  to  indicate  to  him  that 
a  radical  change  is  needed  in  the  field  work  and  mapping  of  cities,  towns, 
and  additions,  not  to  mention  farms  and  other  tracts  of  land  that  it  may  be 
necessary  to  lay  out  and  describe.  So  long  as  fallible  man  is  responsible  for 
the  accuracy  of  surveys,  maps,  and  descriptions  of  properties,  so  long  will 
there  be  errors ;  but  that  it  is  possible  greatly  to  reduce  their  number  by 
proper  regulation  the  writer  is  fully  persuaded.  What  we  have  been  de- 
scribing are  not  maps  at  all,  or  at  most  they  are  very  imperfect  maps,  and 
"what  constitutes  a  map?"  thus  seems  to  be  a  very  pertinent  question. 

A  map  of  a  city,  town,  or  addition,  or  other  tract  of  land,  serving  as  a 
basis  for  the  description  of  property,  should  furnish  all  the  information 


THE   OWNERSHIP   OF   SURVEYS,    ETC.  355 

necessary  for  the  proper  description  and  location  of  the  various  parcels 
shown,  and  also  of  the  whole  piece.  It  should  further  show  the  exact  loca- 
tion of  the  whole  tract  relatively  to  the  lands  immediately  adjoining ; 
particularly  should  this  be  done  when  an  offset  or  angle  in  a  street  line 
occurs.  To  accomplish  these  things,  there  should  appear  on  the  map  the 
following  items : 

(1)  The  lengths  of  all  lines  shown. 

(2)  The  exact  angle  made  by  all  intersecting  lines. 

(3)  The  exact  position  and  character  of  all  monuments  set,  with  notes  of 
reference  points. 

(4)  The  number  of  each  block  and  lot. 

(5)  The  names  of  all  streets,  streams  or  bodies  of  water,  and  recognized 
land  marks. 

(6)  The  scale. 

(7)  The  direction  of  the  meridian  and  a  note  as  to  whether  the  true  or 
magnetic  meridian  is  shown.     (It  should  be  the  true  meridian.) 

(8)  The  angles  of  intersection  made  by  the  lines  of  adjoining  property 
with  the  boundaries  of  the  tract  mapped. 

(9)  The  exact  amount  of  offset  in  lines  that  may  extend  from  the  out- 
side through  the  tract  mapped. 

(10)  A  simple,  complete,  and  explicit  title,  including  the  date  and  the 
name  of  the  surveyor. 

All  this  is  necessary  to  make  the  map  valuable  for  description  and  loca- 
tion of  the  property  it  represents. 

Of  course  monuments  will  not  be  shown  if  none  have  been  set,  and  very 
frequently  none  are  set,  either  from  carelessness  on  the  part  of  the  surveyor, 
or  an  unwillingness  on  the  part  of  the  owner  to  pay  their  cost.  Monuments  of 
a  permanent  character  should  be  set  at  each  corner  of  a  tract  surveyed,  and 
at  least  two,  visible  the  one  from  the  other,  should  be  on  the  line  of  each 
street.  If  these  monuments  are  not  placed  on  the  center  lines  of  the  streets, 
they  should  be  at  uniform  distances  from  the  center  or  property  lines.  If 
placed  with  reference  to  the  center  line  they  should  all  be  on  the  same  side 
of  the  center.  In  streets  extending  east  and  west  the  monuments  should  all 
be  on  the  north  of  the  center,  or  they  should  all  be  on  the  south,  and  at 
uniform  distances.  In  streets  extending  north  and  south  the  monuments 
should  all  be  on  the  east  of  the  center  or  all  on  the  west.  Uniformity  in 
such  practice  saves  a  vast  amount  of  time. 

Monuments  may  be  set  at  uniform  distances  from  the  block  lines,  in  the 
sidewalk  area,  and  this  is  an  excellent  practice.  The  stakes  or  monuments 
set  at  the  corners  of  the  blocks  in  additions,  or  town  sites,  should  never  be 
the  only  stakes  or  monuments  set  in  the  tract. 

That  the  map  may  be  reliable  there  should  appear  on  it  the  following : 

(1)  The  certificate  of  the  surveyor  that  he  has  carefully  surveyed  the 


356  APPENDIX. 

land,  that  the  map  is  a  correct  representation  of  the  tract,  and  that  he  has 
set  monuments  (to  be  described)  at  the  points  indicated  on  the  map. 

(2)  The  acknowledged  signature  of  all  persons  possessing  title  to  any  of 
the  land  shown  in  the  tract,  and,  if  possible,  signatures  of  adjoining  owners. 

(3)  If  of  an  addition,  the  acknowledged  dedication  to  public  use  forever 
of  all  areas  shown  as  streets  or  roads. 

(4)  If  a  street  of  full  width,  whose  center  line  is  a  boundary  of  the  tract, 
is  shown,  the  acknowledged  signature  of  the  owner  of  the  adjoining  property, 
unless  his  half  of  the  street  has  been  previously  dedicated. 

It  has  been  already  stated  that,  in  some  states,  a  map  may  be  filed  at  the 
request  of  any  person,  and  without  signature.  This  practice  frequently 
leads  to  trouble.  The  writer  knows  of  cases  in  which  owners  of  large 
tracts  of  land  have  had  those  tracts  subdivided  and  have  taken  land  of 
adjoining  non-resident  owners  for  street  purposes  without  the  consent  or 
knowledge  of  those  owners.  When,  at  a  later  day,  the  owners  of  the  land 
so  taken  have  objected  and  attempted  to  close  half  of  the  street,  trouble  of 
a  serious  character  has  arisen.  The  same  trouble  has  occurred  where 
streets  have  been  run  through  narrow  gores  of  land  and  have  subsequently 
been  completely  closed,  leaving  houses  built  on  the  mapped  property  with- 
out outlet.  Time  and  again  have  cases  of  this  sort  come  to  the  knowledge 
of  the  writer. 

Having  pointed  out  certain  evils,  it  remains  to  suggest  a  remedy.  It  lies 
in  the  enactment  of  laws  governing  these  matters.  There  should  be  in- 
cluded in  the  statutes  of  every  state  a  law  explicitly  defining  what  shall 
appear  on  every  map  filed  for  reference,  and  making  it  a  misdemeanor  to 
file  a  map  that  does  not  strictly  conform  to  the  requirements.  In  the 
absence  of  such  laws  it  is  believed  that  the  young  surveyor  can  assist 
greatly  in  a  much-needed  reform,  by  following  the  principles  suggested  in 
this  paper  as  the  correct  ones,  and  avoiding  the  errors  here  indicated. 

It  is  hoped  that  those  graduates  of  our  engineering  schools  who  drift  into 
this  class  of  work  will  be  guided  by  a  higher  principle  than  that  which 
actuates  the  surveyor  who  covers  up  his  tracks,  at  the  expense  of  his  em- 
ployer, in  order  to  secure  a  monopoly  of  the  business  of  the  locality.  The 
young  surveyor  can  spend  his  energies  to  greater  advantage  in  devising  new 
and  better  methods  of  work,  than  in  inventing  ways  for  hiding  information 
that  belongs  to  his  employer.  Certainly  a  thorough  education  should  so 
broaden  the  young  man's  views  as  to  make  it  impossible  for  him  to  be 
controlled  by  those  meaner  instincts  which,  if  indulged,  lead  ever  to  narrow 
the  vision  and  prevent  one  from  perceiving  the  greater  problems  that  con- 
tinually present  themselves  for  solution. 


GEOGRAPHICAL  POSITIONS  OF  BASE  LINES,   ETC.      357 


IV.  GEOGRAPHICAL  POSITIONS   OF  BASE  LINES  AND  PRINCI- 
PAL MERIDIANS  GOVERNING  THE   PUBLIC  SURVEYS. 

THE  system  of  rectangular  surveying,  authorized  by  law  May  20,  1785, 
was  first  employed  in  the  survey  of  United  States  public  lands  in  the  state 
of  Ohio. 

The  boundary  line  between  the  states  of  Pennsylvania  and  Ohio,  known 
as  "  Ellicott's  line,"  in  longitude  80°  32'  20"  west  from  Greenwich,  is  the 
meridian  to  which  the  first  surveys  are  referred.  The  townships  east  of  the 
Scioto  River,  in  the  state  of  Ohio,  are  numbered  from  south  to  north,  com- 
mencing with  No.  1  on  the  Ohio  River,  while  the  ranges  are  numbered  from 
east  to  west,  beginning  with  No.  1  on  the  east  boundary  of  the  state,  except 
in  the  tract  designated  "  U.  S.  military  land,"  in  which  the  townships  and 
ranges  are  numbered,  respectively,  from  the  south  and  east  boundaries  of 
said  tract. 

During  the  period  of  one  hundred  and  nine  years  since  the  organization 
of  the  system  of  rectangular  surveying,  numbered  and  locally  named  prin- 
cipal meridians  and  base  lines  have  been  established,  as  follows: 

The  first  principal  meridian  begins  at  the  junction  of  the  Ohio  and  Big 
Miami  rivers,  extends  north  on  the  boundary  line  between  the  states  of 
Ohio  and  Indiana,  and  roughly  approximates  to  the  meridian  of  longitude 
84°  48'  50"  west  from  Greenwich.  The  ranges  of  the  public  surveys  in  the 
state  of  Ohio,  west  of  the  Scioto  River,  are,  in  part,  numbered  from  this 
meridian.  For  further  information  in  regard  to  numbering  of  townships 
and  ranges  of  the  early  surveys  in  Ohio,  the  reader  is  referred  to  the  state 
map  prepared  in  the  General  Land  Office. 

The  second  principal  meridian  coincides  with  86°  28'  of  longitude  west 
from  Greenwich,  starts  from  a  point  two  and  one  half  miles  west  of  the 
confluence  of  the  Little  Blue  and  Ohio  rivers,  runs  north  to  the  northern 
boundary  of  Indiana,  and,  with  the  base  line  in  latitude  38°  28'  20",  governs 
the  surveys  in  Indiana  and  part  of  those  in  Illinois. 

The  third  principal  meridian  begins  at  the  mouth  of  the  Ohio  River  and 
extends  north  to  the  northern  boundary  of  the  state  of  Illinois,  and  with 
the  base  line  in  latitude  38°  28'  20",  governs  the  surveys  in  the  state  east 
of  the  third  principal  meridian,  with  the  exception  of  those  projected  from 
the  second  principal  meridian,  and  the  surveys  on  the  west,  to  the  Illinois 
River.  This  meridian  is  nearly  coincident  with  89°  10'  15"  of  west  longi- 
tude from  Greenwich. 

The  fourth  principal  meridian  begins  at  a  point  on  the  right  bank  of  the 
Illinois  River,  in  latitude  40°  00'  30"  north,  and  longitude  90°  28'  45"  west 
from  Greenwich,  and  with  the  base  line  running  west  from  the  initial  point, 
governs  the  surveys  in  Illinois  west  of  the  Illinois  River  and  west  of  that 
part  of  the  third  principal  meridian  which  lies  north  of  the  river. 

The  fourth  principal  meridian  also  extends  north  through  Wisconsin 
and  northeastern  Minnesota,  and,  with  the  south  boundary  of  Wisconsin  as 


358  APPENDIX. 

its  base  line,  governs  all  the  surveys  in  the  former  and  those  in  the  latter 
state  lying  east  of  the  Mississippi  River,  and  the  third  guide  meridian  west 
(of  the  fifth  principal  meridian  system),  north  of  the  river. 

The  fifth  principal  meridian  starts  from  the  old  mouth  of  the  Arkansas 
River,  and  with  the  base  line  running  west  from  the  old  mouth  of  the 
St.  Francis  River,  governs  the  surveys  in  Arkansas,  Missouri,  Iowa,  North 
Dakota ;  those  in  Minnesota,  west  of  the  Mississippi  River  and  west  of  the 
third  guide  meridian  north  of  the  river ;  and  in  South  Dakota  all  east  of 
the  Missouri  River,  and  the  surveys  on  the  west  side  of  the  river  to  a  limit- 
ing line  following  the  third  guide  meridian  (of  the  sixth  principal  meridian 
system),  White  River,  and  the  west  and  north  boundaries  of  the  Lower  Brule 
Indian  Reservation.  This  meridian  is  nearly  coincident  with  91°  03'  42" 
longitude  west  from  Greenwich. 

The  sixth  principal  meridian,  which  is  approximately  the  meridian  of 
97°  23'  west  longitude  from  Greenwich,  extends  from  the  base  line  coinci- 
dent with  the  north  boundary  of  Kansas  in  latitude  40°  north,  south  through 
the  state  to  its  south  boundary,  in  latitude  37°  north,  and  north  through 
Nebraska  to  the  Missouri  River ;  and  governs  the  surveys  in  Kansas  and 
Nebraska;  the  surveys  in  Wyoming,  except  those  referred  to  the  Wind 
River  meridian  and  base  line,  which  intersect  in  latitude  43°  01'  20"  north, 
and  longitude  108°  48'  40"  west  from  Greenwich ;  the  surveys  in  Colorado, 
except  those  projected  from  the  New  Mexico  and  Ute  meridians,  the  latter 
intersecting  its  base  line  in  latitude  39°  06'  40"  north  and  longitude 
108°  33'  20"  west  from  Greenwich ;  and  the  surveys  in  South  Dakota 
extended,  or  to  be  extended,  over  the  tract  embracing  the  Pine  Ridge  and 
Rosebud  Indian  reservations. 

In  addition  to  the  above-mentioned  numbered  principal  meridians, 
other  principal  meridians  with  local  names  have  been  established  as 
follows : 

The  Michigan  meridian,  in  longitude  84°  22'  24"  west  from  Greenwich, 
with  a  base  line  in  latitude  42°  26'  30"  north  (eight  miles  north  of  Detroit), 
governs  the  surveys  in  Michigan. 

The  Tallahassee  meridian,  in  longitude  84°  16'  42"  west  from  Greenwich, 
runs  north  and  south  from  the  initial  point  on  the  base  line  at  Tallahassee, 
in  latitude  30°  28'  north,  and  governs  the  surveys  in  Florida. 

The  Saint  Stephen's  meridian,  in  longitude  88°  02'  west  from  Greenwich, 
begins  at  the  initial  point  (Ellicott's  corner),  on  the  base  line,  in  latitude 
31°  north,  extends  south  to  Mobile  Bay  and  north  to  latitude  33°  06'  20", 
and  governs  the  surveys  in  the  southern  district  of  Alabama,  and  in  Pearl 
River  district  lying  east  of  the  river  and  south  of  the  Choctaw  base  line,  in 
latitude  31°  52'  40"  north,  in  the  state  of  Mississippi. 

The  Huntsville  meridian  begins  on  the  northern  boundary  of  Alabama, 
in  latitude  34°  59'  north,  longitude  86°  34'  45"  west  from  Greenwich,  extends 
south  to  latitude  33°  06'  20"  north,  and  governs  the  surveys  in  the  northern 
district  of  Alabama. 

The   Choctaw  meridian  begins  on  the  Choctaw  base  line,  latitude    31° 


GEOGRAPHICAL   POSITIONS   OF   BASE   LINES,  ETC.      359 

54'  40"  north,  longitude  90°  14'  45"  west  from  Greenwich,  runs  north  to  the 
south  boundary  of  the  Chickasaw  cession,  in  latitude  34°  19'  40"  north,  and 
governs  the  surveys  east  and  west  of  the  meridian,  and  north  of  the  base 
line. 

The  Chickasaw  meridian  begins  on  the  north  boundary  of  Mississippi 
in  latitude  34°  59'  north,  longitude  89°  15'  west  from  Greenwich,  extends 
south  to  latitude  33°  48'  45"  north,  and  governs  the  surveys  in  north 
Mississippi. 

The  Washington  meridian  begins  on  the  base  line  in  latitude  31°  north, 
longitude  91°  9'  15"  west  from  Greenwich,  extends  north  to  the  Mississippi 
River,  and  governs  the  surveys  in  the  southwestern  angle  of  the  state  of 
Mississippi. 

The  Saint  Helena  meridian  begins  at  the  initial  point  of  the  Washington 
meridian,  in  latitude  31°  north,  and  longitude  91°  09'  15"  west  of  Greenwich, 
extends  south  to  the  Mississippi  River,  and  governs  the  surveys  in  the 
Greensburg  and  southeastern  districts  of  Louisiana,  east  of  the  Mississippi 
River. 

The  Louisiana  meridian,  in  longitude  92°  24'  15"  west  of  Greenwich, 
extends  from  the  Gulf  of  Mexico  to  the  north  boundary  of  Louisiana,  and 
with  the  base  line  through  the  initial  point,  conforming  to  the  parallel  of 
31°  north  latitude,  governs  all  the  surveys  in  the  state  west  of  the  Missis- 
sippi River. 

The  New  Mexico  meridian,  in  longitude  106°  53'  40"  west  from  Green- 
wich, extends  through  the  territory,  and  with  the  base  line,  in  latitude  34° 
15'  25"  north,  governs  the  surveys  in  New  Mexico,  except  those  in  the  north- 
west corner  of  the  territory,  referred  to  Navajo  meridian  and  base  line, 
which  have  their  initial  point  in  latitude  35°  45'  north,  longitude  108°  32' 
45"  west  from  Greenwich. 

The  Salt  Lake  meridian,  in  longitude  111°  54'  00"  west  from  Greenwich, 
has  its  initial  point  at  the  corner  of  Temple  Block,  in  Salt  Lake  City,  Utah, 
extends  north  and  south  through  the  territory,  and,  with  the  base  line, 
through  the  initial,  and  coincident  with  the  parallel  of  40°  46'  04"  north 
latitude,  governs  the  surveys  in  the  territory,  except  those  referred  to  the 
Uintah  meridian  and  base  line  projected  from  an  initial  point  in  latitude  40° 
26'  20"  north,  longitude  109°  57'  30"  west  from  Greenwich. 

The  Boise  meridian,  longitude  116°  24'  15"  west  from  Greenwich,  passes 
through  the  initial  point  established  south  29°  30'  west,  nineteen  miles 
distant  from  Boise  City,  extends  north  and  south  through  the  state,  and, 
with  the  base  line  in  latitude  43°  46'  north,  governs  the  surveys  in  the  state 
of  Idaho. 

The  Mount  Diablo  meridian,  California,  coincides  with  the  meridian  of 
121°  54'  48"  west  from  Greenwich,  intersects  the  base  line  on  the  summit  of 
the  mountain  from  which  it  takes  its  name,  in  latitude  37°  51'  30"  north, 
and  governs  the  surveys  in  the  state  of  Nevada,  and  the  surveys  of  all 
central  and  northern  California,  except  those  belonging  to  the  Humboldt 
meridian  system. 


360  APPENDIX. 

The  Humboldt  meridian,  longitude  124°  08'  west  from  Greenwich,  inter- 
sects the  base  line  on  the  summit  of  Mount  Pierce,  in  latitude  40°  25'  12" 
north,  and  governs  the  surveys  in  the  northwestern  corner  of  California, 
lying  west  of  the  Coast  range  of  mountains,  and  north  of  township  5  south, 
of  the  Humboldt  meridian  system. 

The  San  Bernardino  meridian,  California,  longitude  116°  56'  15"  west 
from  Greenwich,  intersects  the  base  line  on  Mount  San  Bernardino,  latitude 
34°  07'  10"  north,  and  governs  the  surveys  in  southern  California,  lying 
east  of  the  meridian,  and  that  part  of  the  surveys  situated  west  of  it  which 
is  south  of  the  eighth  standard  parallel  south,  of  the  Mount  Diablo  meridian 
system. 

The  Willamette  meridian,  which  is  coincident  with  the  meridian  of  122° 
44'  20"  west  from  Greenwich,  extends  south  from  the  base  line,  in  latitude 
45°  31'  north,  to  the  north  boundary  of  California,  and  north  to  the  inter- 
national boundary,  and  governs  all  the  public  surveys  in  the  states  of 
Oregon  and  Washington. 

The  Black  Hills  meridian,  longitude  104°  03'  west  from  Greenwich,  with 
the  base  line  in  latitude  44°  north,  governs  the  surveys  in  the  state  of  South 
Dakota,  north  and  west  of  White  River,  and  west  of  the  Missouri  River 
(between  latitudes  45°  55'  20"  and  44°  17'  30"),  the  north  and  west  bounda- 
ries of  the  Lower  Bruld  Indian  Reservation,  and  the  west  boundary  of  range 
79  west,  of  the  fifth  principal  meridian  system. 

The  Montana  meridian  extends  north  and  south  from  the  initial  monu- 
ment on  the  summit  of  a  limestone  hill,  eight  hundred  feet  high,  longitude 
111°  38'  50"  west  from  Greenwich,  and  with  the  base  line  on  the  parallel  of 
45°  46'  48"  north  latitude,  governs  the  surveys  in  the  state  of  Montana. 

The  Gila  and  Salt  River  meridian  intersects  the  base  line  on  the  south 
side  of  Gila  River,  opposite  the  mouth  of  Salt  River,  in  latitude  33°  22'  40" 
north,  longitude  112°  17'  25"  west  from  Greenwich,  and  governs  the  surveys 
in  the  territory  of  Arizona. 

The  Indian  meridian,  in  longitude  97°  14'  30"  west  from  Greenwich, 
extends  from  Red  River  to  the  south  boundary  of  Kansas,  and  with  the 
base  line  in  latitude  34°  30'  north,  governs  the  surveys  in  the  Indian  Terri- 
tory, and  in  Oklahoma  Territory  all  surveys  east  of  100°  west  longitude  from 
Greenwich. 

The  Cimarron  meridian,  in  longitude  103°  west  from  Greenwich,  extends 
from  latitude  36°  30'  to  37°  north,  and  with  the  base  line  in  latitude  36° 
30'  north,  governs  the  surveys  in  Oklahoma  Territory  west  of  100°  west 
longitude  from  Greenwich. 


TABLES. 


361 


V.   TABLES. 
TABLE  I. 

CORRECTION  TO  ONE  HUNDRED  UNITS  MEASURED  ALONG  THE 
SLOPES  GIVEN. 


UNITS  RISE  IN  100. 

CORRESPONDING 
VERTICAL  ANGLE. 

CORRECTION. 

1.02 

o°  35' 

0.005 

2.01 

i°  09' 

O.O2O 

3-°3 

i°  44' 

0.046 

4-02 

2°    1  8' 

0.081 

5-01 

2°   52' 

0.125 

6.00 

3°  26' 

0.179 

7.00 

4°  oo' 

0.244 

8.02 

.  4°  35' 

0.320 

9.01 

5°  °9' 

0.404 

10.01 

5°  43' 

0.497 

2O.OI 

n°  19' 

1.617 

30.00 

1  6°  42' 

.  4.218 

4O.OO 

21°  48 

7.151 

50.00 

26°  34' 

10.559 

TABLE   II.1 

CORRECTION  COEFFICIENT  FOR  TEMPERATURE  AND  HYGROMETRIC 

CONDITIONS. 

This  correction  is  used  when  no  hygrometric  observations  have  been  made.  To  the 
difference  in  altitude  found  in  Table  III.  for  the  given  barometer  readings  is  added 
algebraically  the  product  of  that  difference  and  the  correction  below  given,  according 
to  the  formula,  Diff.  Alt.  =  (Diff.  by  Table  III.)  (1  +  c). 


SUM  0.  T.2 

CORR. 
COEFF.3 

SUM  O.  T. 

CORR. 

COEFF. 

SUM  O.  T. 

CORR. 
COEFF. 

0° 

O.IO24 

70° 

0.0273 

I40° 

0.0471 

10 

0.0915 

80 

0.0166 

150 

0.0575 

20 

0.0806 

90 

0.0058 

160 

0.0677 

30 

0.0698 

100 

0.0049 

170 

0.0779 

40 

O.O592 

110 

0.0156 

1  80 

0.0879 

5° 

0.0486 

1  20 

0.0262 

60 

O.OjSo 

130 

0.0368 

1  Computed  from  Tables  I.  and  IV.,  Appendix  10,  "  U.  8.  Coast  Survey  Report "  for  1881. 
*  Sum  of  Observed  Temperatures.  »  Correction  Coefficient. 


362 


APPENDIX. 


TABLE   III.' 
BAROMETRIC   ELEVATIONS. 

Giving  altitudes  above  arbitrary  sea  level  (barometer  reading  30  inches) 
for  various  barometer  readings  B. 

To  determine  difference  of  elevation  of  two  points  having  barometer 
readings  B  and  Bv  take  from  the  table  the  altitudes  corresponding  to  B 
and  .Bj,  and  correct  their  difference  by  Table  II.  The  corrected  difference 
is  the  quantity  required. 


B. 

A. 

DlFF. 
FOR   .01. 

B. 

A. 

DJFF. 

FOR   .01. 

B. 

A. 

DlFF. 
FOR    .01. 

Inches. 

Feet. 

Feet. 

Inches. 

Feet. 

Feet. 

Inches. 

Feet. 

Feet. 

II.  0 

27,336 

—  24.6 

14.0 

20,765 

-19-5 

17.0 

15,476 

—  16.0 

II.  I 

27,090 

24.4 

14.1 

20,570 

"9-3 

17.1 

•5>3l6 

'5-9 

II.  2 

"•3 
ii.  4 

26,846 
26,604 
26,364 

24.2 
24.0 

21.8 

14.2 

14-3 
14.4 

20,377 
20,186 

19,997 

19.1 
18.9 
18.8 

17.2 

J7-3 
17.4 

15,157 

14,999 
14,842 

15.8 

'5-7 
15.6 

11.5 

26,126 

^  J  •" 

23.6 

14-5 

19,809 

18.6 

17-5 

14,686 

I  c.  c 

ii.  6 

25,890 

23-4 

I4.6 

19,623 

18.6 

17.6 

H,53i 

J   J 

11.7 
ii.  8 

25,656 

23.2 

23.0 

14.7 
I4.8 

19,437 
19,252 

18.5 
18.4 

17.7 
17.8 

"4,377 
14,223 

15.4 

11.9 

25,194 

22.8 

14.9 

19,068 

1  8.  2 

17.9 

14,070 

I  c  2 

I2.O 

24,966 

22.6 

15.0 

18,886 

18.1 

18.0 

13,918 

1  ji  * 

12.  1 
12.2 
12.3 

24,740 

24,  5  l6 
24,294 

22.4 
22.2 
22.  1 

I5-I 
15.2 

'5-3 

18,705 
18,525 
18,346 

18.0 
17.9 
17.8 

18.1 
18.2 
18.3 

13,767 
13,617 
13,468 

15.0 
14.9 
14.9 

12.4 

24,073 

21.9 

15-4 

18,168 

17.6 

18.4 

13,319 

14.7 

12-5 
12.6 

23,854 
23,637 

21.7 
21.6 

15-5 
15.6 

17,992 
17,817 

17-5 
17.4 

18.5 
18.6 

13,172 
13,025 

14.7 
14.6 

I2.7 
12.8 

23,421 
23,207 

21.4 
21.2 

15-7 
15-8 

17,643 
17,470 

17-3 
17.2 

18.7 
18.8 

12,879 
12,733 

14.6 

14.4 

12-9 

22,995 

2I.O 

15-9 

17,298 

17.  i 

18.9 

12,589 

14.4 

13.0 

I3-I 
I3.2 

22,785 
22,576 
22,368 

2O-9 
20.8 

20.  6 

16.0 
16.1 
16.2 

17,127 

16,958 
16,789 

16.9 
16.9 
16.8 

19.0 
19.1 
19.2 

12,445 
12,302 
12,160 

14-3 
14.2 
14.2 

13-3 
13-4 

22,162 
21,958 

20.4 
20.  i 

16-3 
16.4 

16,621 
i6,454 

16.7 
16.6 

19-3 
16.4 

12,018 
11,877 

14.1 
14.0 

13-5 
I3.6 

13-7 
13-8 

2I,757 
2I,557 
21,358 

2  1  ,  1  6O 

20.  o 
19.9 
19.8 
19.8 

16.5 
16.6 
16.7 
16.8 

16,288 
16,124 
15,961 
15,798 

16.4 
16.3 
16.3 
16.2 

19-5 
19.6 
19.7 
19.8 

",737 
",598 
",459 
11,321 

13-9 
13-9 
,3.8 
13.7 

13-9 
14.0 

20,962 
20,765 

-19.7 

16.9 
17.0 

15,636 
15,476 

—  16.0 

19.9 

20.  0 

11,184 
11,047 

-'3-7 

i  Taken  from  Appendix  10,  "  U.  S.  Coast  and  Geodetic  Survey  Report "  for  1881 . 


TABLES. 


363 


TABLE   III.    (continued). 


B. 

A. 

DlFF. 

FOR    .01. 

B. 

A. 

DlFF. 
FOR    .01. 

B. 

A. 

DlFF. 
FOR    .01. 

Incites. 

Feet. 

Feet. 

Inches. 

Feet. 

Feet. 

Inches. 

Feet. 

feet. 

20.0 

11,047 

-13.6 

23-7 

6,423 

-II-5 

27.4 

2,470 

—  9-9 

20.1 

10,911 

13.5 

23.8 

6,308 

11.4 

27-5 

2,371 

9-9 

20.2 
20.3 

10,776 
10,642 

13-4 
13.4 

23-9 
24.0 

6,194 
6,080 

11.4 
1  1-3 

27.6 
27.7 

2,272 
2,173 

9-9 
9.8 

20.4 

10,508 

13-3 

24.1 

5,967 

"•3 

27.8 

2,075 

9.8 

20-5 
20.6 
20.7 

I0>375 
10,242 

10,110 

13-3 
13.2 

24.2 

24-3 
24.4 

5,854 
5,741 
5,629 

"•3 

II.  2 
I  I.I 

27.9 
28.0 
28.1 

1,977 
i,  880 

1,783 

9-7 
9-7 
9  7 

20.8 

9,979 

* 

24-5 

5,518 

1  1.  1 

28.2 

1,686 

y  / 
9.7 

20.9 

21.0 

9,848 
9,718 

13.0 
12.9 

24.6 
24.7 

5,296 

II.  I 

I  I.O 

28.3 
28.4 

1,589 
i,493 

y*  / 
9.6 
9.6 

21.  I 

21.2 

9,589 
9,460 

12.9 

1.2.8 

24.8 
24.9 

5,186 
5>°77 

10-9 

28.5 
28.6 

',397 
1,302 

9-5 

Q.C 

21.3 

9,332 

12.8 

25.0 

4,968 

IO-9 

28.7 

1,207 

s   J 

9-5 

21-4 

9,204 

12.7 

25.1 

4,859 

10.8 

28.8 

1,112 

9-4 

2I-5 

9,077 

12.6 

25.2 

4,75' 

10.8 

28.9 

1,018 

9-4 

21.6 

8,951 

12.6 

25-3 

4,643 

10.8 

29.0 

924 

9-4 

21.7 

21.8 

8,825 
8,700 

12.5 

12.5 

25-4 
25-5 

4,535 
4,428 

10.7 
10.7 

29.1 
29.2 

830 
736 

9-4 
9-3 

21.9 

8,575 

12.4 

25.6 

4,32i 

10.6 

29-3 

643 

9-3 

22.O 

8,45  * 

12.4 

25-7 

4,215 

10.6 

294 

55° 

9.2 

22.1 
22.2 

8,327 
8,204 

12.3 

12.2 

25.8 
25-9 

4,109 
4,004 

10.5 
10.5 

29-5 
29.6 

458 
366 

9-2 

9.2 

22.3 
22.4 

8,082 
7,960 

12.2 
12.2 

26.O 
26.1 

3,899 
3,794 

10.5 
10.4 

29.7 
29.8 

274 
182 

9.2 
9.1 

22.5 

7,838 

12.  1 

26.2 

3,690 

10.4 

29.9 

91 

9.1 

22.6 

7,717 

12.0 

26.3 

3,586 

10.3 

30.0 

oo 

9.1 

22.7 
22.8 

7,597 

7,477 

I2.O 
1  1.9 

26.4 
26.5 

3,483 
3,38o 

10.3 
10.3 

3O.I 
30.2 

-91 
181 

9.0 
9.0 

22.9 

7,358 

II.9 

26.6 

3,277 

IO.2 

30.3 

271 

9.0 

23.0 

7,239 

ii.  8 

26.7 

3,175 

IO.2 

30.4 

361 

9.0 

23.1 

7,121 

11.7 

26.8 

3,073 

10.  1 

30-5 

45  * 

8.9 

23.2 

7,004 

1  1.7 

26.9 

2,972 

IO.  I 

30.6 

540 

8.9 

23-3 

6,887 

1      / 
II-7 

27.0 

2,871 

IO.  I 

30-7 

629 

8.8 

23-4 

6,770 

ii.  6 

27.1 

2,770 

IO.O 

30.8 

717 

8.8 

23-5 

6,654 

n.6 

27.2 

2,670 

IO.O 

30.9 

805 

-8.8 

23-6 

6,538 

—  11.5 

27-3 

2,570 

—  IO.O 

3I.O 

-893 

23-7 

6,423 

27.4 

2,470 

364 


APPENDIX. 


TABLE  IV. 
POLAR  DISTANCE  OF  POLARIS.     For  January  1  of  years  named. 


1894 

1897 

1900 

1903 

1906 

1909 

1912 

1915 

1918 

1921 

10  IS43' 

i°  14-5°' 

i°  13-55' 

i°  12.62' 

i°  11.68' 

i°  10.75' 

i°  09.82' 

i°  08.88' 

i°  07.97' 

i°  07.03' 

sin  polar  distance 
Sm  of  azimuth  at  elongation  =    cosine  latitude   • 

Latitude  =  altitude  of  Polaris  at  culmination  ±  polar  distance  — refraction  correc- 
tion given  below. 


LATITUDE. 

CORRECTION,  MINUTES. 

LATITUDE. 

CORRECTION,  MINUTES. 

20° 

3° 
40 

2.60 
I.6S 
I-I3 

50° 
60 

0.80 
o-SS 

TABLE   V.1 

AMOUNT  AND  VARIATION  OF  THE  MAGNETIC  NEEDLE  FROM  ITS 
MEAN  DAILY  POSITION. 

The  letters  E  and  W  indicate  which  side  of  the  mean  position  the  needle  points. 


LOCAL  MEAN  TIME;  MORNING  HOURS. 


6* 

7h 

8» 

9h 

ioh 

Ith 

I2h 

December,  January,  February  : 
Latitude  37°  to  49°    
Latitude  25°  to  37°    
March,  April,  May  : 
Latitude  37°  to  49°         .     . 

i 
0.7  E 
o.iW 

2  6E 

1 
i.iE 
o.iE 

3  8  E 

I 
1.9  E 
i.oE 

4dE 

I 

2.2  E 
2.0  E 

,  e  E 

1 

i-5  E 

2.2  E 
I  2  E 

o.i  W 
i.i  E 

i  6  E 

1.8  W 

0.5  W 

3  8  W 

Latitude  25°  to  37°    
June,  July,  August  : 
Latitude  37°  to  49°    
Latitude  25°  to  37°    
September,  October,  November: 
Latitude  37°  to  49°    
Latitude  25°  to  37°    

1.6  E 

4.0  E 
24  E 

1.8  E 
0.9  E 

2.8  E 

5-6  E 
4.0  E 

2.6  E 

21  E 

3-3  E 

5-7  E 
4.2  E 

3-i  E 

26E 

2.6  E 

4-5  E 
2.9  E 

2.5  E 

21  E 

i.iE 

1.7  E 

0.5  E 

i.oE 
06  E 

0.6  W 

1.6  E 
1.6  W 

i-S  E 
OQ  \V 

1.9  W 

4.1  W 
2.8  W 

3-3  W 

21  W 

SEASON  AND  POSITION  IN  LATITUDE. 

LOCAL  MEAN  TIME;  AFTERNOOM  HOURS. 

oh 

I* 

2h 

3h 

4h 

5h 

6* 

December,  January,  February  : 
Latitude  37°  to  49°    
Latitude  25°  to  37°    
March,  April  .May: 

/ 
1.8  W 
0.5  W 

3.8  W 
1.9  W 

4.1  W 

2.8  W 

3-3  W 

2.1  W 

/ 

2.9  W 
1.5  W 

4.8  W 
2.6  W 

5.6  W 
3.2  W 

4.0  W 
2.3  W 

i 
2.8  W 
1.8  W 

4.6  W 
2.8  W 

5-6  W 

3.1  w 

3-4  W 
1.9  W 

/ 

2.1  W 

1.6  W 

3-8  W 
2.4  W 

4.6  W 
2.4  W 

2.3  W 

1.2  W 

1 
13  W 
i.oW 

2-5  W 
1.6  W 

3.0  W 
i.S  W 

1.2  W 

0.7  W 

/ 

0.7  W 
0.4  W 

14  W 
0.9  W 

14  W 
0.8  W 

0.6  W 
0.4  W 

1 

0.2  W 

o.i  W 

0.7  W 
0.5  W 

0.6  W 
0.4  W 

o.i  W 

0.2  W 

Latitude  25°  to  37°    

June,  July,  August  : 
Latitude  37°  to  49°    . 

Latitude  25°  to  37°    
September,  October,  November  : 
Latitude  37°  to  49°    
Latitude  25°  to  37°    

i  From  "  Manual  of  Instructions  "  issued  by  the  U.  S.  Land  Office  to  Surveyors  General. 


TABLES. 
TABLE   VI 


365 


APPROXIMATE  LOCAL  MEAN  TIMES  (COUNTING  FROM  NOON  24  HOURS) 
OF  THE  ELONGATIONS  AND  CULMINATIONS  OF  POLARIS  IN  THE  YEAR 
1897  FOR  LATITUDE  40°  N. ;  LONGITUDE  6h  W.  FROM  GREENWICH. 


DATE. 

EAST 
ELONGATION. 

WEST                     UPPER 
ELONGATION.         CULMINATION. 

LOWER 
CULMINATION. 

h. 

m. 

h. 

in. 

Jan.     i 

0 

38.2 

12 

27.8 

6 

33-6 

18 

31-6 

v    '5 

23 

39-o 

II 

32.5 

5 

38.6 

17 

36.3 

Feb.     i 

22 

3'-8 

IO 

25-4 

4 

31.2 

16 

29.2 

'5 

21 

36.6 

9 

30.2 

3 

35-9 

15 

33-9 

Mar.    i 

20 

41.4 

8 

34-9 

2 

40.7 

H 

38.7 

15 

19 

46-3 

7 

39-8 

I 

45-7 

13 

43-7 

Apr.    i 

18 

39-3 

6 

32.8 

O 

38.6 

12 

36-7 

15 
May     i 

\l 

44-3 
41-5 

5 
4 

37-8 
35-o 

23 
22 

39-7 
36.8 

II 
10 

41.7 
38.8 

15 

15 

46.6 

3 

40.1 

21 

41.9 

9 

43-9 

June    i 

14 

39-9 

2 

33-4 

2O 

35-3 

8 

37-3 

*5 

13 

45-o 

i 

38.5 

19 

40.4. 

7 

42.4 

July     i 

12 

42.4 

0 

35-9 

18 

37-8 

6 

39-8 

*5 

II 

47-5 

23 

37-i 

17 

42.9 

5 

44-9 

Aug.    i 

IO 

41.0 

22 

30.6 

16 

36.4 

4 

38.4 

«s 

9 

46.1 

21 

35-7 

15 

4i-5 

3 

43-5 

Sept.    i 

8 

39-5 

2O 

29.1 

14 

34-9 

2 

36.9 

15 

7 

44-6 

19 

34-2 

»3 

40.0 

I 

42.0 

Oct.     i 

6 

41.8 

18 

31-4 

12 

37-2 

0 

39-2 

is 

5 

46.8 

17 

36-4 

II 

42.2 

23 

40.3 

Nov.    i 

4 

40.0 

16 

29.6 

IO 

35-4 

22 

33-4 

15 

3 

44-8 

15 

34-4 

9 

40.2 

21 

38-2 

Dec.     i 

2 

41.8 

H 

3i-4 

8 

37-2 

2O 

35-2 

15 

I 

46-5 

13 

36.1 

7 

41.9 

19 

39-9 

To  refer  to  any  calendar  day  other  than  the  first  and  fifteenth  of  each  month, 
subtract  3.94m  for  every  day  between  it  and  the  preceding  tabular  day,  or  add 
3.94m  for  every  day  between  it  and  the  succeeding  tabular  day. 

To  refer  the  tabular  times  to  any  year  subsequent  to  the  year  1897,  add 
0.25m  (nearly)  for  every  additional  year  (after  1900,  0.2m). 

Also,  For  the  second  year  after  a  leap  year,  add,  0.9m. 
For  the  third  year  after  a  leap  year,  add,  1.7m. 
For  leap  year  before  March  1,  add,  2.6m. 

For  leap  year  on  and  after  March  1,  subtract,  1.2m. 

For  the  first  year  after  a  leap  year  the  table  is  correct,  except  for  the  regu- 
lar annual  change. 

To  refer  the  tabular  times  to  other  longitudes  than  six  hours,  add  when 
east,  and  subtract  when  west  of  six  hours,  0.16m  for  each  hour. 

To  refer  to  any  other  than  the  tabular  latitude  between  the  limits  of  25°  and 
50°  north,  add  to  the  time  of  west  elongation  0.13m  for  every  degree  south  of 
latitude  40°,  and  subtract  from  the  time  of  west  elongation  0.18m  for  every  degree 
north  of  40°.  Reverse  these  signs  for  corrections  to  the  times  of  east  elongation. 
For  latitudes  as  high  as  60°,  diminish  the  times  of  icest  elongation  and  increase 
the  times  of  east  elongation  by  0.23m  for  every  degree  north  of  latitude  40°. 

1  Computed  from  information  contained  in  the  "  Manual  of  Instructions  "  issued 
by  the  General  Land  Office.  The  information  was  furnished  by  the  U.  S.  Coast  and 
Geodetic  Survey. 


366  APPENDIX. 

TABLE   VII.1 

REFRACTION  CORRECTIONS  TO  DECLINATION  OF  THE  SUN. 
The  hour  angle  is  the  time  either  side  of  noon. 


LATI- 
TUDE. 

HOUR 

ANGLE. 

DECLINATIONS. 

+  20° 

+*> 

+  10° 

+5° 

0° 

-5° 

—  10° 

-15° 

-~» 

0    • 

b. 

25  oo 

O 

o  05 

0  10 

o  15 

O  21 

o  27 

o  33 

o  40 

o  48 

o  57 

2 

o  08 

o  14 

o  19 

o  25 

o  31 

o  38 

o  46 

o  54 

i  05 

3 

O  12 

o  18 

o  24 

o  30 

o  37 

o  44 

o  53 

i  04 

i  18 

4 

o  23 

o  29 

o  35 

o  45 

o  53 

i  03 

i  if) 

i  31 

1  52 

5 

o  49 

o  59 

I  IO 

i  24 

1  52 

2  07 

2  44 

346 

5  43 

27  30 

o 

2 

o  08 

0  II 

o  13 

O  10 

o  18 

O  22 

S3 

o  30 
o  34 

o  36 
o  41 

o  44 
o  49 

o  52 

I  00 

i  02 

I  10 

3 
4 

o  28. 

O  22 

o  35 

0  28 

o  42 

o  35 
o  50 

o  42 

I  OO 

o  50 
I  II 

I  00 

i  26 

I  II 

i  43 

i  26 

2  09 

5 

o  54 

I  05 

i  18 

i  34 

1  54 

2  24 

3  " 

438 

8  15 

30  oo 

0 

0  10 

o  '5 

0  21 

o  27 

o  33 

o  40 

o  48 

o  57 

i  08 

2 

o  14 

o  19 

o  25 

o  31 

o  38 

o  46 

o  54 

i  18 

3 

o  20 

o  26 

o  32 

o  39 

o  47 

o  55 

i  06 

i  19 

i  36 

4 

o  32 

o  39 

o  46 

o  52 

i  06 

i  19 

i  35 

*  57 

2  29 

5 

I  OO 

I  IO 

I  24 

i  52 

2  07 

2  44 

3  46 

5  43 

13  06 

32  30 

0 

o  13 

o  18 

o  24 

o  30 

o  36 

o  44 

o  52 

i  02 

I  14 

2 

o  17 

O  22 

o  28 

o  35 

o  42 

o  50 

I  OO 

i  ii 

I  26 

3 

o  23 

o  29 

o  35 

o  43 

0  51 

I  01 

I  13 

i  28 

i  47 

4 

o  35 

o  43 

o  51 

I  OI 

i  13 

I  27 

1  46 

2  13 

2  54 

5 

i  03 

i  15 

i  3' 

1  53 

2  20 

3  05 

4  25 

736 

35  °° 

o 

0  15 

0  21 

o  27 

o  33 

o  40 

048 

0  57 

I  08 

I  21 

2 

0  2O 

o  25 

o  32 

o  38 

0  46 

0  55 

1  05 

i  18 

i  35 

3 

o  26 

o  33 

o  39 

o  47 

o  56 

i  07 

I  21 

i  38 

2  00 

4 

o  39 

o  47 

o  56 

i  07 

I  20 

i  36 

1  59 

2  32 

3  25 

• 

5 

i  07 

I  20 

i  38 

2  OO 

2  34 

3  29 

5  *4 

10  16 

1  Computed  from  formula 


57"cot  (&  +  N). 


in  which  8  is  the  declination,  plus  when  north,  and  minus  when  south ;  and  N  an 
auxiliary  angle  found  by 

tan  N  =  cot  <?>  cos  t, 

in  which  <j>  is  the  latitude  of  the  place,  and  t  the  angle  between  the  meridian  of  the 
place  and  the  meridian  through  the  sun  at  the  given  time,  — called  the  "  hour  angle." 
The  formulae  are  from  Chauvenet's  "  Spherical  and  Practical  Astronomy,"  vol.  I., 
p.  171.  The  table  was  computed  by  Mr.  Edward  W.  Arms,  C.E.,  for  Messrs.  W.  & 
L.  E.  Gurley,  of  Troy,  N.Y.,  and  is  here  used  by  their  permission. 


TABLES. 


367 


TABLE   VII.    (continued). 


LATI- 
TUDE. 

HOUR 
ANGLE. 

DECLINATIONS. 

+  20° 

+15° 

-HO- 

+5° 

0° 

-5° 

-10° 

-S° 

—  2O° 

37  30 

o 

o  18 

o  24 

o  30 

o  36 

o  44 

o  52 

I  02 

I  H 

I  29 

2 

0  22 

o  28 

o  35 

o  42 

o  50 

I  OO 

I  12 

I  26 

i  45 

3 

o  29 

o  36 

o  43 

o  52 

I  02 

I  14 

I  29 

i  49 

2  16 

4 

o  43 

o  51 

I  01 

I  13 

I  27 

i  49 

2  14 

2  54 

4  05 

5 

I  II 

I  26 

i  44 

2  IO 

2  49 

3  55 

6  15 

H  58 

40  oo 

0 

0  21 

o  27 

o  33 

o  40 

048 

o  57 

i  08 

I  21 

i  03 

2 

o  25 

o  32 

o  39 

o  46 

o  52 

i  06 

i  19 

i  35 

i  57 

3 

o  33 

o  40 

o  48 

0  57 

I  08 

I  21 

i  38 

2  O2 

2  36 

4 

o  47 

0  55 

i  06 

i  19 

1  36 

I  58 

2  30 

3  21 

4  59 

5 

i  15 

i  3i 

i  5i 

2  2O 

3  05 

4  25 

7  34 

25  18 

42  30 

o 

o  24 

o  30 

o  36 

o  44 

o  52 

i  02 

i  14 

i  29 

i  49 

2 

o  28 

o  35 

o  39 

o  50 

I  00 

I  12 

i  26 

i  45 

2  II 

3 

o  36 

o  43 

o  52 

I  02 

i  13 

I  29 

i  49 

2  17 

2  59 

4 

o  50 

I  OO 

i  ii 

I  26 

I  44 

2  10 

2  49 

3  55 

6  16 

5 

i  16 

1  36 

i  58 

2  30 

3  22 

5  oo 

9  24 

— 

45  °° 

o 

o  27 

o  33 

o  40 

o  48 

o  57 

I  08 

I  21 

i  39 

2  O2 

2 

o  32 

o  39 

o  46 

o  52 

I  O6 

I  19 

i  35 

i  57 

2  29 

3 

o  40 

o  47 

o  56 

I  07 

I  21 

1  38 

2  OO 

2  34 

3  29 

4 

o  54 

i  04 

i  16 

i  33 

I  54 

2  24 

3  ii 

438 

8  15 

5 

i  23 

i  4i 

2  05 

2  41 

3  40 

5  40 

12  O2 

— 

— 

47  30 

o 

o  30 

o  36 

o  44 

o  52 

i  02 

I  14 

I  29 

i  49 

2  18 

2 

o  35 

o  42 

o  50 

I  OO 

I  12 

I  26 

i  45 

2  OI 

2  51 

3 

o  43 

o  51 

I  01 

I  13 

I  28 

i  47 

2  15 

2S6 

408 

4 

o  56 

I  09 

I  23 

I  40 

2  05 

2  40 

3  39 

5  37 

ii  18 

5 

i  27 

1  46 

2  12 

2  52 

4  oi 

6  30 

16  19 

— 

50  oo 

o 

o  33 

o  40 

048 

o  57 

I  08 

I  21 

i  39 

2  02 

236 

2 

o  38 

o  46 

0  55 

i  06 

I  18 

i  35 

i  57 

2  28 

3  '9 

3 

o  47 

o  56 

i  06 

i  19 

i  36 

2  29 

2  31 

3  23 

5  02 

4 

I  02 

I  14 

i  29 

i  48 

2  16 

258 

4  18 

6  59 

19  47 

5 

I  30 

I  51 

2  I9 

3  04 

4  22 

7  28 

24  10 

— 

52  30 

o 

o  36 

o  44 

o  52 

I  02 

I  M 

i  29 

i  49 

2  18 

3  05 

2 

o  43 

o  50 

o  59 

I  II 

I  26 

i  42 

2  23 

2  49 

3  55 

3 

o  50 

I  OO 

i  ii 

I  26 

i  45 

2  II 

2  51 

258 

6  22 

4 

i  05 

i  18 

i  35 

2  IO 

2  28 

3  19 

4  53 

842 

— 

5 

i  34 

i  56 

2  27 

3  16 

4  47 

852 

— 

— 

55  o° 

o 

o  40 

o  48 

0  57 

i  08 

I  21 

i  39 

2  O2 

236 

3  33 

2 

o  46 

o  55 

i  18 

I  34 

i  56 

2  30 

3  15 

4  47 

3 

o  55 

i  06 

i  19 

i  35 

i  58 

2  30 

3  21 

4  58 

9  19 

4 

I  10 

i  23 

i  42 

2  06 

2  43 

3  44 

5  49 

12  4I 

— 

5 

1  37 

2  OI 

2  34 

3  28 

5  '5 

10  18 

— 

368  APPENDIX. 

TABLE  VIII.1 
MAGNETIC  DECLINATION. 

Formulas  giving  approximately  the  magnetic  declination  at  the  places  named 
and  for  any  time  within  the  limits  of  the  period  of  observation.  The  places  are  di- 
vided into  three  groups,  as  follows : 

GROUP  I. — Magnetic  stations  on  the  eastern  coast  of  the  United  States  and  in- 
clusive of  the  region  of  the  Appalachian  range,  with  some  additional  stations  in 
Newfoundland  and  other  foreign  localities. 

GROUP  II.  —  Magnetic  stations  mainly  in  the  central  part  of  the  United  States 
between  the  Appalachian  and  Rocky  Mountain  ranges,  with  additions  in  British 
North  America,  Canada,  the  West  Indies,  and  Central  America. 

GROUP  III.  —  Magnetic  stations  on  the  Pacific  coast  and  Rocky  Mountain  re- 
gion ;  also  in  Mexico  and  Alaska  and  in  some  foreign  countries. 

D  stands  for  declination,  +  indicating  west,  and  —  east  declination ;  m  stands 
for  t  —  1850.0  or  for  the  difference  in  time,  expressed  in  years  and  fraction  of  a  year, 
for  any  time  t  and  the  middle  of  the  century ;  a  *  indicates  uncertainty. 


NAME  OF  STATION 
AND  STATE. 

LATI- 
TUDE. 

WEST 
LONGI- 
TUDE. 

THE  MAGNETIC  DECLINATION 
EXPRESSED  AS  A  FUNCTION  OF  TlME. 

GROUP  I. 

0       ' 

0       ' 

00                                                            0 

Saint    Johns,    New- 

47 34-4 

5241.9 

D=  +  21.94+  8.89  sin  (  i.  05  771  +  63.4)* 

foundland. 

Quebec,  Canada. 

46  48.4 

71  14.5 

D=  +  14.66+  3X>3sin(i.4   771  +  4.6) 

+  0.61  sin  (4.0    771+  0.3) 

Charlottetown.P.E.I. 

4614 

6327 

D=  +  15.95+   7.785111(1.2    771  +  49.8) 

Montreal,  Canada. 

45  30-5 

73  34-6 

D=  +  n.88+  4.i7sin(i.5    771  —  18.5) 

Eastport,  Me. 

4454-4 

6659.2 

D=  +  i5.i8+  3.79  sin(i.  25  771+31.1)* 

Bangor,  Me. 

44  48.2 

68  46.9 

D=  +  13.86+   3.55  sin(i.3om+  8.6) 

Halifax,  Nova  Scotia. 

4439.6 

63  35-3 

D=  +  i6.i8  +  4.53sin(i.o   771+46.1)* 

Burlington,  Vt. 

4428.5 

73  12.0 

D=  +  10.81  +  3.65  sin(i.  30  m  —  20.5) 

+  o.i8sin(7.o   771  +  132) 

Hanover,  N.  H. 

4342-3 

7217.1 

D=+  9.80+  4.O2Sin(i.4   771  —  14.1)* 

Portland,  Me. 

43  38.8 

70  16.6 

D=  +  11.40+   3.28sin(i.3om+   2.7) 

Rutland,  Vt. 

43  36-5 

72  55-5 

P=  +  10.03+  3.82sin(i.5    771-24.3) 

Portsmouth,  N.  H. 

43  04-3 

70  42.5 

D=  +  10.71+  3.36  sin  (i.  44  m-  7.4) 

Chesterfield,  N.  H. 

42  53-5 

7224 

D=+  9.60+  3.843^(1.35  m  —  16.1)* 

Newburyport,  Mass. 

4248.9 

7°  49-2 

D=  +  10.07+  3-Q2sin(i.35  m  —   i.o) 

Williamstown,  Mass. 

4242.8 

73  13-4 

D  =  +   8.84+   3.i3sin(i.4   771—14.0)* 

Albany,  N.  Y. 

42  39-2 

7345-8 

D=+  8.17+  3.02  sin  (i.  44m-  8.3) 

Salem,  Mass. 

4231-9 

7°  52-5 

D=+  9.98+   3.85sin(i.4   m—  5.1)* 

Oxford,  N.  Y. 

42  26.5 

75  40-5 

D=+  6.19+   3.24  sin(i.35  771-18.9) 

Cambridge,  Mass. 

42  22.9 

71  07.7 

D  =  +  9-54+   2.69sin(i.3om+   7.0) 

+  o.i8sin(3.2   771+44) 

Boston,  Mass. 

4221.5 

71  °3-9 

D=+  9.48+   2.94  sin  (i.  3    m+  3.7) 

Provincetown,  Mass. 

4203.1 

7011.3 

D  =  +  9.67+  3.04  sin  (i.  3    7)i+ii.o)* 

Providence,  R.  I. 

41  50.2 

71  23.8 

D=+  9.10+  2.99  sin  (i.  45  m-  3.4) 

+  0.26  sin  (    7    771  +  84) 

Hartford,  Conn. 

4i  45-9 

72  40.4 

D=+  8.06+   2.90  sin(i.  25  m  —  26.4) 

New  Haven,  Conn. 

41  18.5 

D=+   7-78+   3.11  sin(i.4O77i  —  22.1) 

Nantucket,  Mass. 

41  17.0 

70  06.0 

D=  +  8.61+  2.83  sin(i.35  m  +  19.7) 

Cold  Spring  Harbor, 

Long  Island,  N.  Y. 
New  York  City,  N.Y. 

4052 
4042.7 

7328 
7400.4 

D=+  7.19+  2.52  sin(i.  35  m  —  11.4) 
D=+  7.04+  2.77  sin(i.  30771—18.1) 

+  0.14  sin  (6.3    wi  +  64) 

From  Appendix  7,  "  U.  8.  Coast  and  Geodetic  Survey  Report  "  for  1888. 


TABLES. 


369 


NAME  OF  STATION 

LATI- 

WEST 

THE  MAGNETIC  DECLINATION 

AND  STATE. 

TUDE. 

LONGI- 
TUDE. 

EXPRESSED  AS   A   FUNCTION   OF  TlME. 

0       • 

0       • 

00                                                           0 

Bethlehem,  Pa. 

40  36.4 

75  22.9 

D=+   5.40+   3.13  sin(i.55  7/1-38.3) 

Huntingdon,  Pa. 

4031 

7802 

°  =  +  3-76+  2.93sin(i.48m-35.2) 

New  Brunswick,  N.J. 

40  29.9 

74  26.8 

D=+   5.11+  2.94  sin  (1.307/1+  4.2) 

Jamesburg,  N.  J. 

40  21 

7427 

D=+  6.03+  2.94  sin  (i.  40  7/i  -22.4) 

Harrisburg,  Pa. 

40  15.9 

76  52.9 

D=+  2.93+  2.98  sin  (i.  50  m+  0.2) 

Hatboro,  Pa. 

4012 

75  °7 

D=+  5.17+  3.  1  6  sin  (i.  54  7/1-16.7) 

+  0.22  sin  (4.  i    771  +  157) 

Philadelphia,  Pa. 

39  56-9 

7509.0 

D  =  +  5.36+  3.i7sin(i.5om-26.i) 

+  o.i9sin(4.o   7/1+146) 

Chambersburg,  Pa. 

3955 

7740 

D=+  2.79+  3.  10  sin  (i.  55  7/1-30.6) 

+  o.2osin(4.6   7/1+124) 

Baltimore,  Md. 
Washington,  D.  C. 

39  17-8 
38  53-3 

76  37-o 
7700.6 

D  =  +  3.20+  2.57  sin(i.45  m  —  21.2) 
D=+  2.73+  2.5  7  sin  (i.  45  m  -2  1.  6) 

+  o.i4sin(i2    7/4  +  27) 

Cape  Henlopen,  Del. 
Williamsburg,  Va. 

3846.7 
37  1  6.2 

75  05-° 
76  42.4 

D=+  4.01+  3.22  sin  (1.35  7/1-25.2) 
D=+   2.33+   2.56301(1.5    7/1  —  38.1) 

Cape  Henry,  Va. 

7600.4     |D=+  2.42+  2.25  sin(  1.47  7/1  —  30.6) 

Newbern,  N.  C. 

35  °6 

7702 

D  =  +  0.63+   2.56  sin  (1.45  7/i  —  18.2)* 

Milledgeville,  Ga. 

33  04-2 

8312 

D=—  3.10+   2.53  sin(i.4om  —  61.9)* 

Charleston,  S.  C. 

32  46.6 

7955.8     jD=  —   1.82+   2.75  sin(i.4om—  12.1)* 

Savannah,  Ga. 

32  04.9 

81  05.5 

D=—  2.13+   2.55  sin(i.40  7/1-40.5)* 

Paris,  France. 

48  50.2 

2  20.2E 

D=    +  6°.479  +    i6°.oo2  sin  (0.765771 

+  1  1  8°  46'.5   +   [0.85   -  0.35  sin 

(0.69  «)]    sin    [(4.04  +  0.0054  n 

+  .000035  «'2)n] 

St.     George's,    Ber- 

muda 

3223 

6442 

D=+    6.95  +0.0145  771  +  0.00056  ?»2* 

Kiode  Janeiro,  Brazil 

-2254.8 

4309.5 

D=+   2.19+9.91  sin(o.8o  m  —  10.4)* 

GROUP  II. 

York  Factory,  Brit- 

ish North  America 

56  59-9 

92  26 

D  =  +   7.34+16.03  sin  (1.107/1  —  97.9) 

Fort  Albany,  British 

North  America. 

5222 

8238 

D=  +  15.78+   6.95  sin  (i.207/»  -99.6)* 

f  Duluth,  Minn.,  and 

\  Superior  City,  Wis. 

46  45-5 
46  39-9 

92  04.5  1 
92  04.2  / 

D=—  7.70+  2.41  sin  (1.4   7/1  —  120.0)* 

Sault     Ste.     Marie, 

Mich. 

46  29.9 

842O.I 

D=+   1.54+  2.70  sin(i.45  7/1-58.5) 

Pierrepont      Manor, 

N.  Y. 

4344-5 

76  03.0 

D=+  5.95+  3.788^(1.4   7/1-22.2) 

Toronto,  Canada. 

43  39-4 

79  23.5 

D=+  3.60+   2.82sin(i.4   m—  44.7) 

+  0.09  sin  (9.3   7/1+136) 

+  0.08  sin  (19     7/1  +  247) 

Grand  Haven,  Mich. 

43°5-2 

86  12.6 

D=—  4.95+  0.0380  7>i  +0.00  1  20  m'2 

Milwaukee,  Wis. 

43  02.5 

8754-2 

D=—  4.12+   3.60  sin  (  i.  45  m—  64.5)* 

Buffalo,  N.  Y. 

4252.8 

D=+  3.66+  3.478^(1.4  7/1—27.8) 

Detroit,  Mich. 

42  20.0 

83  03.0 

D=-  0.97+  2.2isin(i.5    7/1-15.3) 

Ypsilanti,  Mich. 

42  14 

8338 

D=—   1.20+  3.40  sin  (i.  40  m—  4.1) 

Erie,  Pa. 

4207.8 

8005.4 

D  =  +  2.17+  2.69  dn(  1.5  7/1—27.3) 

Chicago,  111. 

41  50.0 

87  36.8 

D=-  3.77+  2.48sin(i.45?»-62.5) 

Michigan  City,  Ind. 

4i  43-4 

86  544 

D=-   3.23+   2.42  sin  (i.  4   7/1-48.0) 

Cleveland,  0. 

41  30.4 

8141-5 

D=+  0.47+   2.393111(1.30771-14.8) 

Omaha,  Neb. 

4i  15-7 

95  56.5 

D=-  9.30+  3.34  sin  (i.  30  m-  54.7) 

Beaver,  Pa. 

4044 

8020 

D=+   1.41+  2.72  sin  (i.  40  m-  39.6) 

K'AI'D  SURV.  — 24 


370 


APPENDIX. 


NAME  OF  STATION 

LATI- 

WEST 
LONGI- 

THE MAGNETIC  DECLINATION 

AND  STATE. 

TUDE. 

TUDE. 

EXPRESSED  AS  A  FUNCTION   OF  TlME. 

Pittsburg,  Pa. 

40  27.6 

8000.8 

00                                                           0 

D  =  +   1.85+  2.45  sin  (1.45  771—28.4) 

Denver,  Col. 

3945-3 

104  59-5 

D=  —  15.30+  o.o  ii  771+0.0005  ?7i'2 

Marietta,  O. 

3925 

8128 

D  =  +  0.02+   2.89sin(i.4   771  —  40.5) 

Athens,  O. 

8202 

D=—   1.51+  2.63sin(i.4   771  —  24.7) 

Cincinnati,  O. 

39  08.4 

84  25.3 

D=-  2.59+  2.43  sin  (1.42  m  -37.  9) 

Saint  Louis,  Mo. 
Nashville,  Tenn. 

38  38.0 
36  08.9 

9O  12.2 
8648.2 

D=-  5.91+  3.oosin(i.4om-5i.i)* 
D=-  3-57+  3-33  sin(i.35  771-68.5)* 

(Florence,  Ala. 

3447-2 

874L5 

D=-  4.25+   2.33sin(i.3    771-52.8) 

Mobile,  Ala. 

3041.4 

88  02.5 

D=-  4.38+  2.69  sin  (i.  45  731-76.4) 

Pensacola,  Fla. 

30  20.  8 

87  18.3 

D=-  4.40+  3.i6sin(i.4   771-59.4) 

New  Orleans,  La. 

2957-2 

9003.9 

D=—   5.20+   2.98  sin  (1.4077?—  69.8) 

San  Antonio,  Tex. 

29  25.4 

98  29.3 

D=-  7.40+  2.88sin(i.35?7i-8i.8)* 

Key  West,  Fla. 

24  33-5 

8  1  48.5 

D=  —  4.31+  2.86  sin  (  i.  30  m  —  23.9) 

Havana,  Cuba. 

23  09.3 

8221.5 

D=-  4.25-+  2.74  8dn(i.  25  m-  23.3)* 

Kingston,  Port  Royal, 

Jamaica. 

'755-9 

76  50-6 

D=—  3.81+  2.39  sin  (  1.  10  ?7i  —  10.6) 

Barbados,        Caribbee 

Islands. 

1  3  05-7 

59  37-3 

D=—   1.38+  2.84  sin(  1.  10771+09.4) 

Panama,  Colombia. 

79  32-2 

D=—  5.66+   2.22  sin(i.iom  —  27.8) 

GROUP  III. 

Acapulco,  Mexico. 

1  6  50.5 

99  52-3 

D=-  4.48+  4.4isin(i.o   771-85.7)* 

Vera  Cruz,  Mexico. 

1911.9 

9608.8 

D=-   5.09+  4.22sin(i.2   771-63.4)* 

City  of  Mexico,  Mex. 

19  26.0 

9911.6 

D=-  5-34+   3-28sin(i.o   7/1-87.9)* 

San  Bias,  Mex. 

21  32.5 

105  18.4 

D=-  5.21+  4.26  sin(i.  15  771-96.5)    ' 

San      Lucas,      Lower 

Cal. 

22  53-3 

109  54.7 

D=  —  5.94+  3.68sin(i.2om-  116.8)* 

Mazdalena  Bay,  Lower 

Cal. 

24  38-4 

1  1  2  08.9 

D=—  6.33+  4.17  sin(i.  15  m  —  119.2)* 

Cerros   Island,    Lower 

Cal. 

2804 

115  12 

D=—  7.40+  4.61  sin  (1.05  731—107.0) 

El  Paso,  Tex. 

3i  45-5 

1  06  27.0 

D  =  —  9.08+   3.40  sin(  1.3    731—108.4) 

San  Diego,  Cal. 

3242.1 

"7  »4-3 

D=  —  10.32+  3.00  sin(  i.  lorn—  126.5) 

Santa  Barbara,  Cal. 

34  24-2 

11943.0 

D=  —  11.52+  3.32  sin(i.  10  m—  123.1) 

Monterey,  Cal. 

3636-1 

121  53.6 

D=  —  13.25+  2.83  sin  (1.10771—144.0) 

San  Francisco,  Cal. 
Cape  Mendocino,  Cal. 

37  47-5 
40  26.3 

12227.3 
12424.3 

D=-  13.94+  2.655^(1.05771-135.5) 
D=  —  15.25  +  2.45  sin  (  1.  10771  —  128.0)* 

Salt  Lake  City,  Utah. 

4046.1 

III  53.8 

D=  —  12.40+  4.255^(1.4   771  —  121.6)* 

Vancouver,  Wash. 

45  37-5 

12239.7 

D  =  -i7.93+  3.12  sin(i.35m-  134.1)* 

Walla  Walla,  Wash. 

4604 

II822 

D=  —  17.80+  3.3osin(i.3   731—129.0)* 

Cape  Disappointment, 
Wash. 

46  16.7 

12402.8 

D=-  19-39  +  2.54  sin(i.25  m-  158.7) 

Seattle,  Wash. 

47  35-9 

12220.0 

D=  —  19.19+   3.143111(1.4    731—136.1)* 

Port  Townsend,  Wash. 

48  07.0 

12244.9 

D=-  18.84+   3.00  sin  (i.  45  m  -i  22.1) 

Neah  Bay,  Wash. 

48  21.8 

12438.0 

D=  —  19.83+  2.91  sin(i.4O  771  —  141.6) 

Nootka,  Vancouver  Isl. 

49  35-5 

12637.5 

D=  -21.25+  2.74  sin(i.30  m-  152.0)* 

Captain's  and  Iliuliuk 

Harbors. 

53  52-6 

16631.5 

D=-  18.01  +   i.82sin(i.3    771-69.6)* 

Sitka,  Alaska. 
St.  Paul,  Kadiak  Island. 

57°2-9 
5748.0 

135  '9-7 
15221.3 

D=—  25.79+  3.30  sin  (1.30  TO  —  104.2) 

[)=  -22.21+    5.l8sin(l.3573l-72.5) 

Port  Mulgrave,  Alaska. 

59  33-7 

13945-9 

[)=—  24.03+  7.  77  sin(  i.  30  771—85.8) 

Port  Etches,  Alaska. 
Port  Clarence,  Alaska. 

60  20.7 
65  16 

146  37.6 
1  66  50 

D=-  23.71  +   7.89  sin  (i.  35731-80.9) 
D  =  —  18.98+   7.99801(1.3    771  —  68.4)* 

Dhamisso  Isl.,  Alaska. 

6613 

161  49 

[)=  —  23.62+  7.64  sin  (1.3   771  —  64.0)* 

Petropaulovsk,  Siberia. 

5301 

201  17 

D  =  -  3.35+  2.97'sin(i.3   771+12.2) 

TABLES. 


371 


TABLE  IX. 
ANGULAR  CONVERGENCES  AND  DISTANCES  BETWEEN  MERIDIANS. 

1.  Angular  convergence  of  meridians  per  mile  of  easting  or  westing. 

2.  Distance  between  meridians  converging  by  one  minute. 


LATI- 
TUDE. 

ANGULAR  CON- 
VERGENCE PER 
MILE. 
MINUTES. 

DISTANCE  FOR 
CONVERGENCE 

OFl'. 

FEET. 

LATI- 
TUDE. 

ANGULAR  CON- 
VERGENCE PER 
MILE. 
MINUTES. 

DISTANCE  FOE 
CONVERGENCE 

OF  1'. 

FEET. 

o 

0 

I 

0.015 

34S733 

31 

0.521 

10140 

2 

.030 

I743H 

32 

•542 

975» 

3 

•045 

116150 

33 

.563 

9382 

4 

.061 

87052 

34 

•585 

9034 

5 

.076 

69578 

35 

.607 

8703 

6 

.091 

57917 

36 

.630 

8387 

7 

.107 

49578 

37 

•653 

8087 

8 

.122 

43337 

38 

.677 

7801 

9 

•137 

38436 

39 

.702 

7527 

10 

•'53 

34525 

40 

.727 

7265 

ii 

.169 

31320 

4i 

•753 

7013 

12 

.184 

28642 

42 

.780 

6770 

13 

.200 

26371 

43 

.808 

6537 

14 

.216 

24419 

44 

.836 

6313 

15 

.232 

22723 

45 

.866 

6097 

16 

.249 

21234 

46 

.897 

5888 

17 

.265 

19916 

47 

.929 

5686 

18 

.282 

18740 

48 

.962 

549» 

'9 

.299 

17685 

49 

,998 

53°i 

20 

.316 

16731 

5° 

.032 

5"8 

21 

•333 

15864 

51 

.069 

4940 

22 

•35° 

I5°73 

52 

.108 

4766 

23 

.368 

14348 

53 

.149 

4597 

24 

.386 

13680 

54 

.191 

4433 

25 

.404 

13062 

55 

.236 

4271 

26 

•423 

12488 

56 

.283 

4"5 

27 

.442 

"955 

57 

•333 

3962 

28 

.461 

"457 

58 

•385 

38i3 

29 

.480 

10990 

59 

1.440 

3666 

3° 

.500 

'°552 

60 

1:499 

3523 

372 


APPENDIX. 


TABLE   X.1 

LENGTH  OF  ONE  MINUTE  OF  LATITUDE  AND  ONE  MINUTE  OF  LON- 
GITUDE TO  THE  NEAREST  WHOLE  FOOT. 


LATI- 
TUDE. 

1'  LATITUDE. 
FEET. 

1'  LONGITUDE. 
FEET. 

LATI- 
TUDE. 

1'  LATITUDE. 
FEET. 

1'  LONGITUDE. 
FEET. 

I 

6046 

6086 

3i 

6062 

5222 

2 

6046 

6083 

32 

6063 

5167 

3 

6046 

6079 

33 

6064 

5110 

4 

6046 

6072 

34 

6065 

5°52 

5 

6046 

6064 

35 

6066 

4992 

6 

6047 

6054 

36 

6067 

4930 

7 

6047 

6042 

37 

6068 

4867 

8 

6047 

6028 

38 

6069 

4803 

9 

6047 

6013 

39 

6070 

4737 

10 

6048 

5995 

40 

6071 

4670 

ii 

6048 

5976 

4i 

6072 

4601 

12 

6049 

5955 

42 

6074 

453i 

13 

6049 

5932 

43 

6075 

4459 

H 

6050 

5908 

44 

6076 

4386 

15 

6050 

5881 

45 

6077 

43" 

16 

6051 

5853 

46 

6078 

4236 

17 

6051 

5823 

47 

6079 

4159 

18 

6052 

579i 

48 

6080 

4081 

19 

6052 

5758 

49 

6081 

4001 

20 

6053 

5722 

50 

6082 

3921 

21 

6054 

5685 

51 

6083 

3839 

22 

6055 

5647 

52 

6084 

3756 

23 

6055 

5606 

53 

6085 

3671 

24 

6056 

5564 

54 

6086 

3586 

25 

6057 

5520 

55 

6087 

3499 

26 

6058 

5475 

56 

6088 

3412 

27 

6059 

5427 

57 

6089 

3323 

28 

6059 

5379 

58 

6090 

3234 

29 

6060 

5328 

59 

6091 

3"43 

30 

6061 

5276 

60 

6092 

3051 

1  Abbreviated  from  the  Smithsonian  Geographical  Tables. 


TABLES. 


373 


TABLE   XL 
TRIGONOMETRIC  FUNCTIONS  AND  FORMULAS.     SOLUTION  OF  TRIANGLES. 

By  definition,  if  R  =  1,  p O        c 

ED  =  sine  a. 
OD  -  cosine  a. 
DA  =  versed  sine  a. 
HF  =  coversed  sine  a. 
BA  =  tangent  a. 
jPC  =  cotangent  a. 
OB  =  secant  a. 
OC  =  cosecant «. 

If  R  is  other  than  1,  it  follows  from 
the  above  definitions  and  the  propor- 
tionality of  similar  figures,  that 

5.  BA  =  R  tan  a. 

6.  FC  =  Rcota. 

7.  OB  =  R  sec  a. 

8.  OC  =  R  cosec  a. 


2.  OD  =  Rcosa. 

3.  DA  =  R  versin  a. 

4.  HF  =  R  co versin  a. 


from  which  also  in  any  right  triangle  of  angles  a  and  ft,  if  o  be  the  side 
opposite  the  angle  a,  a  the  side  adjacent  thereto,  and  h  the  hypotenuse, 


9.       sin  a  =  -  =  cos  ft. 
h 

10.       cos  a  =  -  =  sin  ft. 

h, 

11.      tana  =  -  =  cot  £. 
12.      cot  a  =  -  =  tan  fl. 

13.          sec  a  =  -  =  cosec  ft. 
14.      cosec  a  =  -  =  sec  ft. 
15.       vers  a  =            =  covers  /3. 
16.  covers  a  =  —  ^  =  vers  /?. 

Hence, 

So  =  A  sin  a  =  h  cos  5. 
ft=     o     =     o 
sin  a     cos  ft 
!a  =  h  cos  a  =  h  sin  /?. 
f*r»s  ft.       si  n  ^ 


19. 


sin/3 
o  =  a  tan  a  =  a  cot  /J. 

~~  tan  «  ~  cot  y8' 


20. 


±2. 


a  =  o  cot  a  =  o  tan  ft. 

o  =    a     =     a 
cot  a     tan  /?' 

h  =  a  sec  a  =  a  cosec  ft. 


__     __. 
sec  a     cosec  /3 

h  =  o  cosec  a  =  o  sec  /J 

*      =      h 
cosec  a     sec  /?' 


374 


APPENDIX. 


23.  o  =  VA2  _  a2  _  V(A  +  a)(A  -  a). 

24.  a  =  VA2  -  o2  =  V(A  +  o) (A  -  0). 

25.  A  =  Vo2  +  a2. 
26. 


Oblique  triangles  may  be  solved  by  some  one  of  the  following  formulas 

GIVEN.          I       SOUGHT.  FORMULAS. 

27.   A,  B,  a,        |    C,  b,  c,  C  =  180°- (A  +  J5),  0  =    ."     sin  B, 


sin  4 
c          a      sin  (  4    1   5") 

28.   A,  a,  b, 

29.    C,  a,  6, 
30. 
31. 

32. 

5,  C,  c, 
i  (4  +  B), 

A,B, 
c, 

sin  B  =  ?HLd&,    C  =  180°  -  (A  +  B), 
a 

c  =  —  -  —  sin  C. 
sin  A 

$(A  +  B)  =  90°  -  J  C. 

tan  J  (^4  —  .B)  =  a  ~    tan  $(A  +  B). 
a  +  b 

B  =  %(A  +  B)  -  $  (A  -  B). 
j\cosK^4  +  -B) 

/s'nSKl+?) 

33. 
34.    a,  0,  c, 

Area, 

Area  =  \  ab  sin  C. 
If  s  =  i  (a  4  0  +  c), 

sin  i  4  —  *  /&2-Z  ^)(s  ~~  c^ 

V      0C 

' 

tan!  4       ./C*  -»)(*-«) 

V      .(.-a) 

2  Vs(s  —  a)(s  —  ft)(s  —  c) 

be 
ver3    4      2(s-o)(s-c) 

35. 
36.   A,  B,  C,  a, 

Area, 
Area, 

Area  =  Vs(s  -  a)(s  -  o)(.v  -  c). 
,.            a2  sin  B  sin  C 

2  sin  4 

TABLES.  375 

From  the  definitions  of  the  trigonometric  functions,  the  geometrical 
properties  of  right  triangles  and  in  some  cases  algebraic  transformations,  it 
may  be  shown  that  if  A  is  any  angle  and  B  any  other  angle, 

37.  sin.9  A  +cos2-4  =  1. 

38.  sin  A  = — - — -  =  Vl  —  cos'2  A  =  tan  A  cos  A 

vers  A  cot  |  A 


sec  A 


sin  2  A          sin  2  A 


=  cos2^  -sin2  £.4  = 

40.    tan  A  =  sin  A  _     1 
cos  A      cot  A 


1  +  cos  2  A 
=  cot  £  .4  (sec  ,4  -1). 

41.  cot  A  =  °2Ld.  =  _L_  =  Vcosec2  A  -  1 

sin  A      tan  A 

sin  2  A      _  sin  2  A  _  1  +  cos  2  J.  _  tan  ^  A 
~  1  -  cos  2  ^4  ~  vers  2  A  ~     sin  2  ^4      ~  sec  A  -  1 

42.  vers  A  =  1  -cos  ^4  =  sin  A  tan  J  A  =  2  sin2  J  ^4  =  cos  ^4  (sec  ,4  - 1). 

43.  sin  (A  ±  £)  =  sin  ^4  cos  J5  ±  sin  B  cos  .4 . 

44.  cos  (A  ±  B)  =  cos  A  cos  5  =F  sin  /I  sin  5. 


48.     cos  2  4  =  2  cos2 ,4  -  1  =  cos2  A  -  sin2  A  =  1  -  2sin2^. 

tan.4 
1  +  sec  A 


49.      tanM=-^^-  =  cosec^-cot^  =  1-C09^ 


50. 


1  -  tan2  £  4 
51. 


sin  A         cosec  A  —  cot  ^4 


376 


APPENDIX. 


52.   cot   2  A  = 


cot2 .4-1 


53.   vers  A  A  = 


1  -  cos  ^4 


2+V2(l 


vers  A 
1  +  VI  -  £  vers  J. 

54.  vers  2  A  =2  sin2 .4. 

55.  sin  A  +  sin  B  =  2  sin  J  (/I  +  B)  cos  £  (4.  —  -B). 

56.  sin  ^4  —  sin  B  =  2  cos  J  (J.  +  -B)  sin  £  (^4  —  B). 

58.  cos   B  —  cos  A  =2  sin  £  (4.  +  B)  sin  £  ( J.  —  B) . 

59.  sin2  A  -  sin2B  =  cos2B  -  cos2 .4  =  sin  (A  +  B)  sin  (A 

60.  cos2 ,4  -  sin2B  =  cos(J.  +  B)cos(4  -  B). 

61.  tan^+tanB  =  sin(f  +  B\ 

cos  A  cos  B 

62.  tan  A  —  tan  B  = 


cos  ^4  cos  £ 


TABLE  XII. 

LENGTHS  OF  CIRCULAR  ARCS  OF  RADIUS  1,  AND  VARIOUS 
CIRCULAR  MEASURES. 


No. 

DEGREES. 

MINUTES. 

SECONDS. 

No. 

DEGREES. 

MINUTES. 

SECONDS. 

i 

•0174533 

.0002909 

.0000048 

6 

.1047198 

.0017453 

.0000291 

2 

.0349066 

.0005818 

.0000097 

7 

.1221730 

.0020362 

•0000339 

3 

•0523599 

.0008727 

.0000145 

8 

.1396263 

.0023271 

.0000388 

4 

.0698132 

.0011636 

.0000194 

9 

.1570796 

.0026180 

.0000436 

5 

.0872665 

.0014544 

.0000242 

10 

•1745329 

.0029089 

.0000485 

Degrees  in  arc  of  length  equal  to  radius,    57.°  295  780. 
Degrees  in  arc  of  length  equal  to  IT,  180.°  000  000. 

Circumference  =  2  irr  =  360.°  000  000. 

Area  =  Trr2. 


If  I  =  length  of  circular  arc 
d  —  number  of  degrees  in  same 
r  =  radius  of  same 
c  =  chord  of  same 
m  =  middle  ordinate 


180° 

7T     ' 


"180" 
Area  of  sector  =  \  Ir. 

Area  of  sector  =  — -  Trr2. 
o60 

Approximate  area  of  segment  =  |  cm. 


TABLES. 


377 


TABLE   XIII. 
LINEAR  TRANSFORMATIONS. 
1.    Gunter's  Chains  to  Feet. 


CHAINS. 

0.0 

0.01 

.02 

•03 

.04 

•05 

.06 

.07 

.08 

.09 

0.0 

.66 

1.32 

I.98 

2.64 

3-30 

3-96 

4.62 

5.28 

5-94 

.1 

6.60 

7.26 

7.92 

8.58 

9.24 

9.90 

10.56 

11.22 

u.88 

12.54 

.2 

13.20 

13.86 

14.52 

I5.I8 

15.84 

16.50 

17.16 

17.82 

18.48 

19.14 

•3 

19.80 

20.46 

21.12 

21.78 

22.44 

23.10 

23.76 

24.42 

25.08 

25-74 

•4 

26.40 

27.06 

27.72 

28.38 

29.04 

29.70 

30-36 

31.02 

31.68 

32.34 

•5 

33-oo 

33-66 

34-32 

34.98 

35.64 

36-30 

36-96 

37.62 

38.28 

38.94 

.6 

39.60 

40.26 

40.92 

41.58 

42.24 

42.90 

43-56 

44.22 

44.88 

45-54 

•7 

46.20 

46.86 

47-52 

48.18 

48.84 

49.50 

50.16 

50.82 

51.48 

52.14 

.8 

52.80 

53-46 

54.12 

54.78 

55-44 

56.10 

56.76 

57-42 

58.08 

58.74 

•9 

59-40 

60.06 

60.72 

61.38 

62.04 

62.70 

63-36 

64.02 

64.68 

65.34 

0.0 

I.O 

2.O 

3-0 

4.0 

5-o 

6.0 

7.0 

8.0 

9-0 

o.o 

66 

132 

198 

264 

330 

396 

462 

528 

594 

10.0 

660 

726 

792 

858 

924 

990 

1056 

1  1  22 

1188 

1254 

20.0 

1320 

1386 

1452 

1518 

1584 

1650 

1716 

1782 

1848 

1914 

30.0 

1980 

2046 

2112 

2178 

2244 

2310 

2376 

2442 

2508 

2574 

4O.O 

2640 

2706 

2772 

2838 

2904 

297° 

3036 

3102 

3168 

3234 

5O.O 

3300 

3366 

3432 

3498 

3564 

3630 

3696 

3762 

3828 

3894 

60.0 

3960 

4026 

4092 

4158 

4224 

4290 

4356 

4422 

4488 

4554 

70.0 

4620 

4686 

4752 

4818 

4884 

495° 

5016 

5082 

5H8 

5214 

80.0 

5280 

5346 

5412 

5478 

5544 

5610 

5676 

5742 

5808 

5874 

378 


APPENDIX. 


«     ro   u->   f^   ON    »« 


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£3  ftS&S-gi&S  ? 

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i        ^OO\O     f^iOsvO     CS     S?*^ 


M     coi^r^d\w     roint^cf\ 


o   i  ij  5  $<£  5:^  o, 


O\O 
N"^M 

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OO     O 


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gqqqoqqoq 


- 

.    o    —    N    f*i"->vO    r-.oo    O*O 
eoooooooooooooooooo    ON 
•    c5    4<o  od    o    c5    ^-vo  06 


w    00      l^« 


Q     ^     toi^^OCO     ON"-" 
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CM        Tj-    NO 


.     O     S'c^rOTfvO     t^OO     ONO 

S    ^"tTf'1'T'1"<t't1"10 

'cJ'i-NdoodN^j-vdoo 


N     ^t^OOO     O     N     Tj-vO 


l-l       CS1       fO      T}-      *J~t     t~*   OO 

(S      CM      M      M      M      CM      (S 
CM      •"t  \O    00      O      CM      Tj- 


«      N 

fOON 
VO     M 


O     O     O     p'     Q     O 


TABLES. 


379 


3.    Feet  to  Meters. 


1 

1 

0 

I 

2 

3 

4 

5 

6 

.    7 

8 

9 

FEET. 

Meters. 

Meters. 

Meters. 

Meters. 

Meters. 

Meters. 

Meters. 

Meters. 

Meters. 

Meters. 

O 

o.ooo 

0.305 

0.610 

0.914 

I.2I9 

1.524 

1.829 

2.134 

2.438 

2-743 

IO 

3.048 

3-353 

3.658 

3.962 

4.267 

4-572 

4.877 

5.182 

5-486 

5-791 

2O 

6.036 

6.401 

6.706 

7-010 

7-3I5 

7.620 

7.925 

8.229 

8-534 

8.839 

30 

9.144 

9-449 

9-753 

10.058 

10.363 

10.668 

10.972 

11.277 

11.582 

11.887 

40 

12.192 

12.496 

12.801 

13.106 

13.411 

13.716 

14.020 

14.325 

14.630 

14-935 

50 

15-239 

15-544 

15.849 

16.154 

16.459 

16.763 

17.068 

17-373 

17.678 

I7-983 

60 

18.287 

18.592 

18.897 

19.202 

I9-507 

19.811 

2O.II6 

20.421 

20.726 

21.031 

70 

21.335 

21.640 

21-945 

22.250 

22-555 

22.859 

23.164 

23469 

23-774 

24.079 

80 

24.383 

24.688 

24.993 

25.298 

25.602 

25-907 

26.212 

26.517 

26.822 

27.126 

90 

27-431 

27.736 

28.041 

28.346 

28.651 

28.955 

29.260 

29-565 

29.870 

30.174 

100 

30-479 

30-784 

31-089 

31-394 

31-698 

32.003 

32.308 

32.613 

32.918 

33-222 

| 

4.   Meters  to  Feet. 


O 

I 

2 

3 

4 

5 

6 

7 

8 

9 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

0 

O.oo 

3-28 

6.56 

9-84 

13.12 

16.40 

19.69 

22.97 

26.25 

29-53 

IO 

32.81 

36.09 

39-37 

42.65 

45-93 

49-21 

52.49 

55-78 

59-06 

62.34 

20 

65.62 

68.90 

72.18 

75-46 

78.74 

82.02 

85-30 

88.58 

91.87 

95-'5 

30 

98.43 

101.71 

104.99 

108.27 

"i-SS 

114.83 

Ii8.ii 

121.39 

124.67 

127.96 

40 

131.24 

I34-52 

137.80 

141.08 

144.36 

147.64 

150.92 

154.20 

157.48 

160.76 

5° 

164.04 

167.33 

170.61 

I73-89 

177.17 

180.45 

183-73 

187.01 

190.29 

193-57 

60 

196.85 

200.13 

203.42 

206.70 

209.98 

213.26 

216.54 

219.82 

223.10 

226.38 

70 

229.66 

232.94 

236.22 

239-5  I 

242.79 

246.07 

249-35 

252.63 

255.9I 

259.19 

80 

262.47 

265-75 

269.03 

272.31 

275.60 

278.88 

282.16 

285.44 

288.72 

292.00 

90 

295.28 

298.56 

391-84 

305.12 

308.40 

311.69 

3H-97 

318.25 

321-53 

324-81 

100 

328.09 

331-37 

334-65 

337-93 

341.21 

344-49 

347-78 

351.06 

354-34 

357-62 

1 

statute  mile  =  1.6093  kilometers 
kilometer     =  0.6214  statute  miles 


380 


APPENDIX. 


multiplying  the  quantities 
y  2.04.  Use  a  slide  rule. 


ON. 

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ultip 


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22&H 

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TABLES. 


381 


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382 


TABLES   XV,    XVI. 

COMMON   LOGARITHMS  OF  NUMBERS. 
LOGARITHMS  OF  TRIGONOMETRIC  FUNCTIONS, 


EDITED   BY   C.   W.   CROCKETT. 


NOTE. 

THE  well-known  tables  of  Gauss,  Hotiel,  Becker,  and 
Albrecht  have  been  taken  as  the  standards,  and  the  figures 
compared  with  the  more  extensive  tables,  the  doubtful  cases 
being  recomputed. 


384 


EXPLANATION    OF   THE    TABLES 

INTRODUCTORY. 

1.  When  we  have  a  number  with  six  or  more  decimal  places,  and  \ve 
wish  to  use  only  five  : 

(a)  If  the  sixth  and  following  figures  of  the  decimal  are  less  than 
5  in  the  sixth  place,  they  are  dropped;  thus,  0.46437  4999  is  called 
0.46437. 

(6)  If  the  sixth  and  following  figures  of  the  decimal  are  greater  than 
5  in  the  sixth  place,  the  fifth  place  is  increased  by  unity  and  the  sixth 
and  following  places  are  dropped  ;  thus,  0.46437  5001  is  called  0.46438. 

(c)  If  the  sixth  figure  of  the  decimal  is  5,  and  if  it  is  followed  only  by 
zeros,  make  the  fifth  figure  the  nearest  even  figure  ;  thus,  0.46437  500  is 
called  0.46438,  while  0.46438  500  is  also  called  0.46438.  The  number 
is  thus  increased  when  the  fifth  figure  is  odd,  and  decreased  when  it  is 
even,  the  two  operations  tending  to  neutralize  each  other  in  a  series  of 
computations,  and  hence  to  diminish  the  resultant  error. 

2.  Hence  any  number  obtained  according  to  Art.  i  may  be  in  error 
by  half  a  unit  in  the  fifth  decimal  place. 

3.  When  the  last  figure  of  a  number  in  these  tables  is  5,  the  number 
printed  is  too  large,  the  5  having  been  obtained  according  to  Art.  i  (If) ; 
if  the  5   is  without  the   minus  sign,  the  number  printed  is  too  small, 
the  figures  following  the  5  having  been  dropped  according  to  Art.  i  (#). 

4.  The  marginal  tables  contain  the  products  of  the  numbers  at  the 
top  of  the  columns  by  i,  2,  3,  •••  9  tenths,  and  may  be  used  in  multiply- 
ing and  dividing  in  interpolation. 

(a)  To  multiply  38  by  .746  :  38 


38  x. 7=  =26.6  3 

38  x  .4=  15-2;  .'.  38  X  .04    =    1.52 

38  x  .6  =  22.8  ;   .-.  38  x  .006  =      .228  6 


38  X  .746  =  28.348 


3-8 
7.6 
11.4 

'5-2 
19.0 


22.8 
26.6 
3°-4 

9  !  34-2 


In  multiplying  by  the  second  figure  (hundredths),  the  decimal  point 
in  the  table  is  moved  one  place  to  the  left ;  in  multiplying  by  the  third 
(thousandths),  two  to  the  left ;  and  so  on. 


(385) 


386 


EXPLANATION    OF    THE   TABLES. 


To  divide  28  by  38  : 


Dividend, 
Next  less, 

Remainder, 
Next  less, 

Remainder, 

Nearest, 

.'.  Quotient, 


28 
26.6 

i  4 
i  1.4 

26 


corresponding  to  .7 
corresponding  to  .03 


2  6.6  corresponding  to  .007 
.737 


38 

3.8 
7-6 
11.4 

22!8 

26.6 
30.4 
9    34-2 


to  the  nearest  third  decimal  place.     The  decimal  point  is  moved  one 
place  to  the  right  in  each  remainder,  since  the  next  figure  in  the  quotient 
will  be  one  place  farther  to  the  right. 
To  divide  23  by  38  : 


Dividend,      23 
22. 


corresponding  to  .6 


o.o      corresponding  to  .00 

2  o. 
Nearest,  i  9.0  corresponding  to  .005 

.-.  Quotient,  .605 

The  computer  should  use  the  marginal  tables  mentally. 


LOGARITHMS. 

5.  The  logarithm  of  a  number  is  the  exponent  of  the  power  to  which 
a  given  number  called  the  base  must  be  raised  to  produce  the  first 
number.  If  A  =  ea,  a  is  called  the  logarithm  of  the  number  A  to  the 
base  <?,  written  log,  A  =  a. 

e.  If  A  =  ea,  and  B  =  eh,  or  (omitting  subscripts)  \ogA-a,  and 
log  B  =  b,  we  have 

Ax£  =  e*^b;    .-.  \og(A  x  B}  =  a  +  b  ;  .-.  log(A  x£)  =log  ^  +  log B. 
A^B  =  e"-1;    .-.  \og(A  -=-  B)  =  a  —  b  ;   .-.  log(^~^)  =log  A  — 
An         —  e"a ;      .-.  log(^4")         =  na  ;        /.  log(^4")        =;/log^. 

^flA      =  eT'a ;     .-.  log\/ A         =-a;       .-.  log  -\/A        =-\ogA. 


EXPLANATION    OF   THE    TABLES. 


387 


7.  When  the  base  is  not  specified,  it  is  generally  understood  that 
logarithms  to  the  base  10,  or  common  logarithms,  are  meant.  In  this 
system,  since 


o.ooi  = 


=  — „  =  io 


1000      io3 


log  o.ooi  =  —  3 


O.OI      = = -=IO~2,  logO.OI      =—2 

100  10- 

I  I 

O.I         = = =  IO      .  log  O.I        = —   I 

JO  10 

i.                                    ion,  log         1=0 

IO.                                                    IO1,  log         IO  =  +  I 

IOO.                                                       IO2,  log       IOO  =  +  2 

1000.                                    io3,  log  1000  = +  3 


8.  The  logarithm  of  a  number  between  100  and  1000  will  be  a  num- 
ber between  2  and  3,  or  2  -f-  m  where  m  will  be  a  decimal  called  the 
mantissa,  the  integral  portion  of  the  logarithm  being  the  characteristic. 
The  mantissa  is  always  considered  positive;    thus  log  0.002  will  be  a 
number  between  —  2  and   —  3,  that  is,  either  —  3  -f-  m  or  —  2  —  ;;/', 
the  first  form  being  used.     We  write  log  0.002  =  3.30103,  the  negative 
sign  being  placed  over  the  characteristic  to  show  that  the  characteristic 
alone  is  negative. 

9.  Since 

log  (A  x  10")  =  log  A  +  log  ion  =  log  A  -f  n  log  io  =  log  ^4  +  n, 
and  log  (A  -~-  ion)  =  log  A  —  log  io"  =  log  A  —  n  log  io  =  log^  —  n, 
we  have,  if  log  37.3=  1.57171, 


Iog373-  =2.57171, 
Iog373°  =  3-57I7I> 
log  37300  =  4.5  71  71, 


and 
and 
an<3 


log  3.73  =0.57171 
logo.373  =1.57171 
^0.0373  =  2.57171 


Hence  the  position  of  the  decimal  point  affects  the  characteristic  alone, 
the  mantissa  being  always  the  same  for  the  same  sequence  of  figures. 
For  this  reason  the  common  system  of  logarithms  is  used  in  practice. 

10.  The    characteristic   is    found   as  follows  :      When  the  number  is 
greater  than  i,  the  characteristic  is  positive,  ami  is  one  less  than  the  num- 
ber of  digits  to  the  left  of  the  decimal  point  ;  when  the  number  is  less 
tlian  i,  the  characteristic  is  negative,  and  is  one  more  t/ian  t/ie  number 
of  zeros  between  the  decimal  point  and  the  first  significant  figure. 

11.  To  avoid  the  use  of  negative  characteristics  we  add  io  to  the 
characteristic  and  write  —  io  after  the  mantissa,  i.e.  adding  and  subtract- 
ing the  same  quantity,  io.     Thus  log  0.2  ='1.30103  would  be  written 


388  EXPLANATION    OF   THE   TABLES. 

9.30103  —  10.  The  —  10  is  often  omitted  for  brevity  when  there  is  no 
danger  of  confusion,  but  its  existence  must  not  be  forgotten.  Such 
logarithms  are  called  augmented  logarithms. 

In  this  case  the  characteristic  of  the  logarithm  of  a  pure  decimal  is  9 
diminished  by  the  -number  of  ciphers  to  the  left  of  the  first  significant 
figure.  Thus  the  characteristic  of  log  0.004  >s  9  —  2,  or  7>  and  that  of 
log  0.94  is  9  —  o,  or  9. 

12.  The  arithmetical  complement  of  the  logarithm   (written  colog} 
of  a  number  is  the  logarithm  of  its  reciprocal,  and  is  found  by  subtract- 
ing each  figure  of  the  logarithm  from  9,  commencing  at  the  left,  except 
the   last   significant  figure  on  the  right,  which  is  subtracted  from   10. 

For  log-  =  —  log.r  =  10  —  log.r  —  10; 

thus,  if  log x=  2.46403,  colog x=  7-53597—  I0 ; 

if  log  .#  =  8.43000  —  10,  colog  x  =  1.57000. 

TABLE   XV. 

13.  Page  397  contains  the  logarithms  of  numbers  from  i  to  100,  to 
five  decimal  places. 

Pages  398-415  contain  the  mantissas  of  the  logarithms  of  numbers 
from  1000  to  10009,  to  five  decimal  places. 

Pages  416,417,  contain  the  mantissas  of  the  logarithms  of  numbers 
from  10000  to  11009,  to  seven  decimal  places. 

NOTE. — The  mantissas  of  the  logarithms  of  numbers,  except  those  of  the  integral 
powers  of  10,  are  incommensurable,  the  mantissas  in  the  tables  being  found  as 
shown  in  Art.  i. 

To  find  the  Logarithm  of  a  Number. 

14.  The  characteristic  is  found  by  the  rules  in  Arts.  10  and  n,  and 
the  mantissa  from  the  tables,  as  shown  in  Arts.  15,  16,  17,  18. 

15.  When  the  number  has  four  figures.  —  Find  on  pages  398-415 
the  first  three  figures  in  the  column  marked  N,  and  the  fourth  at  the  top 
of  one  of  the  other  columns.     The  last  three  figures  of  the  mantissa  are 
found  in  this  column  on  the  horizontal  line  through   the   first   three 
figures  of  the  given  number  in  column  N.     The  first  two  figures  of  the 
mantissa  are  those  under  L  in  the  same  line  with  the  number,  or  else 
those  nearest  above  it,  unless  the  last  three  figures  of  the  mantissa  as 
given  in  the  tables  are  preceded  by  a  *,  when  the  first  two  figures  are 
found  under  L  in  the  first  line  below  the  number.     Thus  (page  398), 

log  1 136  =  3.05538;  log  1137  =  3.05576;  log  1 138  =  3.05614; 
log  1370  =  3.13672  ;  log  1371  =  3.13704  ;  log  1372  =  3.13735  ; 
log  1380  =  3.13988  ;  log  1381  =  3.14019  ;  log  1382  =  3.14051. 


EXPLANATION    OF    THE   TABLES.  389 

16.  When  the  number  has  less  than  four  figures,  annex  ciphers  on 
the  right  and  proceed  as  in  Art.  15.     Thus, 

log  1.13  =  0.05308  ;  log  12.8  =  1.10721  ;  log  130  =  2.11394  ; 
log  15     =1.17609;  log  1 6     =1.20412;  log  17    =1.2304^. 

17.  When  the  number  has  more  than  four  figures,  as    11.4672. — 
Since  the  mantissa  is  independent  of  the  position  of  the  decimal  point, 
point  off  the  first  four  figures  and   find  the   mantissa  of  log  1146. 72. 
This  will  be  between  the  mantissas  of  log  1146  and  log  1147.     Hence 
find  from  the  tables  the  mantissas  corresponding  to    1146  and  1147; 
multiply  the  difference  between  them  (called  the  tabular  difference)  b£ 
.72,  and  add  the    product   (called   the   correction)   to    log  11.46;  the 
result  will  be  the  logarithm  required. 

Mantissa  of  log  1146  =  05918  log  11.46  =  1.05918 

Mantissa  of  log  1147  =05956  correction  =  38  x  .72  =  27-36 

Tabular  difference      =        38  .-.  log  11.4672  =  1.05945  36 

or  =  1.05945 

XOTK.  —  Since  any  mantissa  in  the  tables  may  be  in  erroj  by  half  a  unit  in  the 
fifth  decimal  place  (Art.  2),  no  advantage  is  gained  by  using  the  sixth  place  in 
the  interpolated  logarithm.  Thus,  according  to  Art.  I,  we  drop  the  .36,  and  call 
log  1 1. 4672=  1.05945. 

NOTE.  — The  marginal  tables  should  be  used  in  multiplying  the  tabular  difference 
to  find  the  correction  (Art.  4). 

NOTE.  — It  is  assumed  that  the  change  in  the  mantissa  is  proportional  to  that  in 
the  number,  as  the  latter  increases  from  1146  to  1147.  An  increase  of  I  in  the  num- 
ber causes  an  increase  of  38  in  the  mantissa;  hence  an  increase  of  .72  in  the  number 
will  cause  an  increase  of  38  X  .72  in  the  mantissa. 

NOTE. — We  could  also  find  the  mantissa  of  log  11.4672  by  subtracting  the  prod- 
uct of  the  tabular  difference  by  .28  (or  i.oo  —  .72)  from  the  mantissa  corresponding 
to  1147;  that  is,  the  required  mantissa  is  05956  — (38x  .28)^05956— 10.64  — 05945 
as  before. 

18.  The  general  rule   is :    Find  the   mantissa   corresponding   to  the 
first  four  figures  of  the  number;  multiply  the  tabular  difference  by  the 
fifth  and  following  figures  treated  as  a  decimal;  and  add  the  product 
to  the  mantissa  just  found. 

The  tabular  difference  is  the  difference  between  the  mantissas  corre- 
sponding to  the  two  numbers  in  the  tables,  between  which  the  given 
number  lies. 

log  1.62163  =  0.20995  ;  log 0.38024  =  1.58006  ;  log 0.085 2 763  =  2.93083  ; 
log  189. 524  =  2. 27767  ;  Iogo.386o2=l. 58661  ;  ^0.0085938  =  3.93419  ; 
log  19983.4  =  4.30067  ;  Iog3-98743  =  o. 60070  ;  logo. 090046  =2.95446. 


390  EXPLANATION    OF   THE    TABLES. 

NOTE.  — Page  397  is  used  when  the  number  contains  less  than  three  figures,  the 
number  being  found  in  the  column  N,  and  the  logarithm  in  the  column  headed  Log. 
The  characteristic  is  given  for  whole  numbers,  and  must  be  changed  for  decimals. 

NOTE. — When  a  number  is  composed  of  three  figures,  find  on  pages  398-415 
the  number  in  the  column  N,  and  the  mantissa  corresponding  in  the  column  L.  o. 

To  find  the  Number  corresponding  to  a  Given  Logarithm. 

19.  From  the  tables  we  find  the  sequence  of  figures  corresponding 
to  the  given  mantissa,  as  shown  in  Arts.  20,  21,  and  22,  the  position  of 
the  decimal  point  being  determined  by  the  characteristic  (Arts.  10,  n). 

20.  When  the  given  mantissa  can  be  foimd  in  the  tables.  —  Find  on 
pages  398-415  the  first  two  figures  of  the  mantissa  under  L  in  the  column 
headed  L.  o.     The  last  three  figures  of  the  mantissa  are  then  sought  for 
in  the  columns  headed  o,  i,  2,  •••  9,  in  the  same  line  with  the  first  two 
figures,  or  in  one  of  the  lines  just  below,  or  in  the  line  next  above 
(where  they  would  be  preceded  by  a  *).     The  first  three  figures  of  the 
required  number  will  be  found  in  the  column  headed  N,  in  the  same 
horizontal  line  with  the  last  three  figures  of  the  mantissa,  and  the  fourth 
figure  of  the  number  at  the  top  of  the  column  in  which  the  last  three 
figures  of  the  mantissa  are  found.     Thus  (page  398), 

0.06221  =  log  1.154  ;  0.06558  —  log  1.163  i  0.06893  =  log  1.172  ; 
0.07004  =  log  1.175  >  0.07188  =  log  1.180  ;  0.08063  =  log  1.204. 

21.  When  the  given  mantissa  can  not  be  found  in  the  tables.  —  If  we 
wish  to  find  the  number  whose  logarithm  is  2.16531,  we  enter  the  tables 
with  16531,  and  find  that  it  lies  between  16524  and  16554,  so  that  the 
given  mantissa  corresponds  to  a  number  between  1463  and  1464.     Also 
16531  exceeds  16524  by  7,  and  this  difference,  divided  by  the  tabular 
difference  30,  gives  .23«"  as  the  amount  by  which  the  required  number 
exceeds  1463.     Hence  2.16531  =  log  I46.323---,  which  we  call  146.32, 
according  to  Art.  i,  the  incompleteness  of  the  tables  making  the  sixth 
figure  uncertain. 

NOTE. — The  marginal  tables  should  be  used  in  dividing  the  difference  between 
the  given  mantissa  and  the  one  next  less  in  the  tables  by  the  tabular  difference. 

22.  The  general  rule  is :    Find  the    number  corresponding   to    the 
mantissa  in  the  tables  next  less  than  the  given  mantissa  ;  divide  the 
excess  of  the  given  mantissa  over  the  one  found  in  the  tables  by  the 
tabular  difference;  and  annex    the   quotient  to    the  first  four  figures 
already  found. 

The  tabular  difference  is  the  difference  between  the  two  mantissas  in 
the  tables,  between  which  the  given  mantissa  lies. 

T.i66oo  =  log  0.14656  ;    0.18002  =  ^1.5136;    2.18200  =  log  152.06  ; 
1.19000  =  log  15.488  ;      4.19680  =  log  15773  ;     1.20020  =  log  15.856. 

23.  For  the  use  of  the  numbers  S',  T',  S",  T",  see  Arts.  35-38. 


EXPLANATION  OF  THE  TABLES.          391 


TABLE  XVI. 

24.  This  table  (pages  420-464)  contains  the  logarithms,  to  five  deci- 
mal places,  of  the  trigonometric  sines,  cosines,  tangents,  and  cotangents 
of  angles   from  o°   to  90°,  for  each  minute.     The   logarithms  in    the 
columns  headed  L.  Sin,  L.  Tan,  and  L.  Cos,  are  augmented,  and  should 
be  diminished  by  10  (Art.   n),  while    those   in   the   columns   headed 
L.  Cot  are  correctly  given. 

25.  Since  secx  = ,  and  cosec.#  =  - — ,  the  logarithms  of  the 

cos  x  sin* 

secant  and  cosecant  of  an  angle  are  the  arithmetical  complements  of 
those  of  the  cosine  and  sine  respectively  (Art.  12). 


To  find  the  Logarithmic  Functions  of  an  Angle  Less  than  90°. 

26.  When  the  angle  is  less  than  45°,  the  degrees  are  found  at  the  top 
of  the  page,  and  the  minutes  on  the  left.     The  numbers  in  the  same 
horizontal  line  with  the  minutes  of  the  angle  are  the  logarithmic  functions 
indicated  by  the  notation  at  the  top  of  the  columns.      Thus  (page  428), 

log  sin  8°  4'  =  9.14714  —  10,         log  tan  8°  4'  =  9.15145  —  10, 
log  cot  8°  4'  =  0.84855,  log  cos  8°  4'  =  9.99568  —  10. 

27.  When  the  angle  is  greater  than  45°,  the  degrees  are  found  at  the 
bottom  of  the  page,  and  the  minutes  on  the  right.     The  numbers  in  the 
same  horizontal  line  with  the  minutes  of  the  angle  are  the  logarithmic 
functions  indicated  by  the  notation  at  the  bottom  of  the  columns.     Thus 
(page  428), 

log  sin  81°  25'  =  9.99511  —  10,     log  tan  81°  25'  =  0.82120, 

log  cot  81°  25'  =  9.17880  —  10,     log  cos  81°  25'  =  9.1  7391  —  10. 

28.  When  the  angle  is  given  to  decimals  of  a  minute.  —  In  finding 
log  sin  30°  8 '.48,    for   example,   we    see    that   it   will   lie   between    the 
logarithmic   sines   of   30°  8'  and  30°  9',  that  is,  between  9.70072  and 
9.70093,  their  difference  21  being  the  change  in  the  logarithmic  sine 
caused  by  a  change  of  i'  in  the  angle.     Hence,  to  find  the  correction 
to  log  sin  30°  8'  that  would  give  log  sin  30°  8'-48  we  multiply  21  by  .48 
(Art.  4).     The  product  10.08  added  to  log  sin  30°  8',  since  log  sin  30°  9' 
is  greater  than  log  sin  30°  8',  gives  log  sin  30°  8'.48  =  9.70082    (Art.  i). 
Similarly,    log  tan  30°  8'.48  =  9.76391,    log  cot  30°  8'-48  =  0.23609,  log 
cos  30°  8'.48  =  9.93691,   the   correction   being   subtracted    in   the   last 
two  cases,  since   the  cotangent  and  the  cosine  decrease  as  the  angle 
increases. 


392  EXPLANATION    OF    THE    TABLES. 

29.  The  general  rule  is :  Find  the  function  corresponding  to  the  given 
degrees  and  minutes;  multiply  the  tabular  difference  by  the  decimals  of 
a  minute ;  add  the  product  to  the  function  corresponding  to  the  given 
degrees  and  minutes  when  finding  the  logarithmic  sine  or  tangent,  and 
subtract  it  when  finding  the  logarithmic  cosine  or  cotangent. 

The  tabular  differences  are  given  in  the  columns  headed  d.  and  c.  d., 
the  latter  containing  the  common  difference  for  the  L.  Tan  and  L.  Cot 
columns.  The  difference  to  be  used  is  that  between  the  functions  cor- 
responding to  the  two  angles  between  which  the  given  angle  lies. 

For  30°  39'.38  :   log  sin  =  9.70747  ;  log  cos  =  9.93462  ; 

log  tan  =  9.77285  ;  log  cot  —  0.22715. 
For  59°  43'-46  :  log  sin  =  9.93632  ;  log  cos  =  9.7025  7  ; 

log  tan  =  0.233 75  ;  log  cot  =  9.76625. 

30.  When  the  angle  is  given  to  seconds,  the  correction  may  be  found 
by  multiplying  the  tabular   difference  by  the  number  of  seconds,  and 
dividing  the  product  by  60. 

To  find  the  Acute  Angle  corresponding  to  a  Given  Logarithmic 
Function. 

31.  The  column  headed  L.  Sin  is  marked  L.  Cos  at  the  bottom,  while 
that  headed  Z.  Cos  is  marked  L.  Sin  at  the  bottom  ;  hence,  if  a  logarith- 
mic sine  or  cosine  were  given,  we  should  expect  to  find  it  in  one  of  these 
two  columns.     Similarly,  we  should  expect  to  find  a  given  logarithmic 
tangent  or  cotangent  in  one  of  the  two  columns  headed  Z.  Tan  and 
Z.  Cot. 

32.  When  the  function  can  be  found  in  the  tables. —  If  a  logarithmic 
sine  is  given,  find  it  in  one  of  the  two  columns  marked  Z.  Sin  and 
L.  Cos ;  if  found  in  the  column  headed  Z.  Sin,  the  degrees  are  taken 
from  the  top,  and  the  minutes  from  the  left  of  the  page  ;  if  in  the 
column  headed  Z.  Cos  but  marked  Z.  Sin  at  the  bottom,  the  degrees 
are  taken  from  the  bottom,  and  the  minutes  from  the  right  of  the  page. 
Thus, 

9. 701 15  =  log  sin  30°  10';  9.9345 7  =  log  sin  59°  20'; 

9.93 724  =  log  cos  30°    4';  9.70590  =  log  cos  59°  28'; 

9. 76406  =  log  tan  30°    9';  0.23130  =  log  tan  59°  35'; 

0.23420  =  log  cot  30°  15';  9.76870  =  log  cot  59°  35'. 

33.  When  the  function  can  not  be  found  in  the  tables.  —  If  we  wish  to 
find  the  angle  whose  logarithmic  sine  is  9.70170,  we  see  on  page  450 
that  the  given  logarithmic  sine  lies  between  9.70159  and  9.70180,  and 


EXPLANATION    OF   THE    TABLES.  393 

hence  the  angle  is  between  30°  12'  and  30°  13'.  The  given  logarithmic 
sine  differs  from  log  sin  30°  12'  by  11,  and  this  difference,  divided  by 
the  tabular  difference  21,  gives  .52+  as  the  decimal  of  a  minute  by 
which  the  angle  exceeds  30°  12'.  Hence  9.70170  =  log  sin  30°  i2'-52, 
which  we  call  30°  12'.$,  since  the  incompleteness  of  the  tables  (Art.  i) 
makes  the  hundredths  of  a  minute  uncertain. 

34.  The  rule  is :  For  a  logarithmic  sine  or  tangent  find  the  degrees 
and  minutes  corresponding  to  the  function  in  the  tables  next  less  than 
the  given  function  ;  divide  the  difference  between  the  given  function  and 
the  one  next  less  by  the  tabular  difference ;  and  the  quotient  will  be  the 
decimal  of  a  minute  to  be  added  to  the  degrees  and   minutes  already 
found.     For  a  logarithmic  cosine  or  cotangent  find  the  degrees  and  min- 
utes corresponding  to  the  function  next  greater  than  the  given  function, 
since  tJte  cosine  and  cotangent  decrease  as  the  angle  increases,  and  divide 
the  difference  between  the  given  function  and  the  one  next  greater  by  the 
tabular  difference,  to  find  the  decimal  of  a  minute. 

The  tabular  difference  is  the  difference  between  the  two  functions  in 
the  tables,  between  which  the  given  function  lies. 

9.70000  =  log  sin  30°  4 '.7;  9. 93500  =  log  sin  59°  25'.;; 
9.93400  =  log  cos  30°  4y'.6  ;  9.70500  =  log  cos  59°  ^'.2  • 
9.77000  —  log  tan  30°  29 '.5  ;  0.23200  =  log  tan  59°  3 7 '.4  ; 
0.23300  =  log  cot  30°  19'.!  ;  9.76400  =  log  cot  59°  51  '.2. 

Angles  Near  o°  or  90°. 

35.  The  assumption  that  the  variations  in  the  functions  are  propor- 
tional to  the  variations  in  the  angles  if  the  latter  are  less  than  i '  fails 
when  the  angle  is  small,  shown  by  the  rapid  changes  in   the   tabular 
differences  on  pages  420,  421,  and  422. 

36.  The  quantities  S'  and  T'  which  are  used  in  this  case  are  defined 
by  the  equations 

0,      ,       sin  <i 

5'==  be > 

a' 

7"=logtan/'-, 
a 

where  «'  is  the  number  of  minutes  in  the  angle.  Their  values  from 
o°  to  i°  40'  (=100')  are  given  at  the  bottom  of  pages  397-415  ;  from 
i°4o'  to  3°  20'  at  the  left  margin  of  pages  398  and  399,  the  first  three 
figures  being  found  at  the  top ;  and  from  3°  to  5°  on  page  418.  Thus, 

for          i'=      i'   (page  399),  S'  =  6.46  373,   T'  =  6. 46  373; 

for        15'=    15'  (page  399),  £'  =  6.46372,   T' =  6.46  373; 

for  2°  40'=  T 60'  (page  399),  S'  =6.46  35  7,   T' =  6.46  404; 

for  4°  20' =  260'  (page  418),  £'  =  6.46331,   7"  =  6.46  456. 
Each  of  these  numbers  should  have  —  10  written  after  it  (Art.  n). 


394  EXPLANATION    OF    THE    TABLES. 

NOTE.  —  The  logarithmic  cosine  of  a  small  angle  is  found  by  the  ordinary  method. 
The  cotangent  of  an  angle  is  the  reciprocal  of  the  tangent,  and  hence  the  logarithmic 
cotangent  is  the  arithmetical  complement  of  the  logarithmic  tangent.  The  formulas 
for  finding  the  logarithmic  cosine,  tangent,  and  cotangent  of  angles  near  90°  are 
given  on  page  419. 

37.  To  find  the  logarithmic  sine  or  tangent  of  a  small  angle.  —  From 
Art.  36,  we  have 

log  sin  a  =  S'  +  log  a', 

log  tan  «  =  T'  +  log  «'. 

Hence,  to  find  the  logarithmic  sine  or  tangent  of  an  angle  less  than 
5°,  find  the  value  of  the  S'  or  T1  corresponding  to  the  angle,  interpolat- 
ing if  necessary,  and  add  it  to  the  logarithm  of  the  number  of  minutes 
in  the  angle. 

Find  log  sin  o°42'.6.     Since  the  angle  is  nearer  43'  than  42',  we  take 

S'  =  6.46  371 
log  42. 6  =  1.62  941 

.•.  log  sin  o°  42  '.6  =  8.09  3 1 2 

Find  log  tan  i°53'.2.  Since  the  angle  is  nearer  i°53r  (=  113')  than 
114',  we  take 

T'  =  6.46  388 
log  113.2  =  2.05  385 

.-.  log  tan  i°53'.2  =  8.51  773 

NOTE. — When  the  angle  is  given  in  seconds,  either  reduce  the  seconds  to  deci- 
mals of  a  minute,  or  use  the  values  of  S"  and  T"  given  at  the  bottom  of  pages 
397-417  and  on  page  418.  They  are  defined  by  the  equations 

c(/       ,       sin  o         ,   „-,,,       ,       tan  a 

S  "  =  log ,  and  T"  —  log , 

a  a 

where  a"  is  the  number  of  seconds  in  the  angle.     Hence 

log  sin  o  =  S"  +  log  a",  and  log  tan  a  =  T"  +  log  a". 

38.  To  find  the  small  angle  corresponding  to  a  given  logarithmic  sine 
or  tangent.  —  From  Art.  36, 

log  «'  =  log  sin  a  —  S', 
log  a'  =  log  tan  «  —  T', 

or  log  «'  =  log  sin  a  +  cpl  .S ', 

log  «'  =  log  tan  a  -\-  cpl  T'. 

When  the  angle  is  less  than  3°,  find  on  pages  420-422  the  value  of 
cpl  S'  (or  cpl  7")  corresponding  to  the  function,  interpolating  if  neces- 
sary, and  add  it  to  log  sin  «  (or  log  tan  «) ;  the  sum  will  be  the  loga- 
rithm of  the  number  of  minutes  in  the  angle. 

In  finding  the  angle  whose  logarithmic  sine  is  8.09006,  we  see  from 


EXPLANATION    OF    THE    TABLES. 


395 


the  L.  Sin  column  (page  420)  that  the  angle  is  between  o°  42'  and  o°  43', 
and  that  the  value  of  cpl  S'  must  be  either  3.53628  or  3.53629.  The 
given  logarithmic  sine  is  nearer  that  of  42'  than  that  of  43';  hence  we 
take 

cpl  5' =3.53628 
log  sin  a  =  8.09006 

log  a' =  1.62634     .'.  «'—  42'. 300. 

When  the  angle  is  between  3°  and  5°,  we  may  find  S'  and  T'  from 
page  418  after  finding  the  angle  approximately  from  pages  423  and  424. 
Thus  in  finding  the  angle  whose  logarithmic  tangent  is  8.77237  we 
find  from  page  423  that  the  angle  is  between  3°  23' and  3°  24',  being 
nearer  3°  23'.  Then  on  page  418  we  have 

*       T'=  6.46423 
log  tan  a  =  8.77237 
.-.  log  tan  a  —  T' =  log  «'==  2.30814    .-.  «'=  2O3'-3O  =  3°  23'. 30. 


Angles   Greater  than  90". 

39.  To  find  the  logarithmic  sine,  cosine,  tangent,  or  cotangent  of  an 
angle  greater  than  90°,  subtract  from  the  given  angle  the  largest  multi- 
ple of  90°  contained  therein.  If  this  multiple  is  even,  find  from  the 
tables  the  logarithmic  sine,  cosine,  tangent,  or  cotangent  of  the  remain- 
ing acute  angle.  If  the  multiple  is  odd,  the  logarithmic  cosine,  sine, 
cotangent,  or  tangent,  respectively,  of  the  remaining  acute  angle  will  be 
the  function  required  ;  thus,  sin  120°  =  sin  (90°  +  30°)  =  cos  30°. 


1.   QUADRANT. 

11.  QUADRANT. 

III.   QUADRANT. 

IV.   QUADRANT. 

jr  = 

a 

90°  + 

180°  +  a 

270°  +  a 

sin  x  = 

+  sin  a 

+  cos  a 

—  sin  a 

—  cos  a 

cos  x  = 

-f  cos  a 

—  sin  a 

—  cos  a 

+  sin  a 

tan  x  = 

+  tan  a 

—  cot  a 

+  tan  a 

—  cot  a 

cot  x  = 

+  cot  a 

—  tan  a 

+  cot  a 

—  tan  a 

Or  we  could  find  the  difference  between  the  angle  and  180°  or  360°, 
and  find   from  the  tables  the  same  function   of  the  remaining  acute 

angle  ;  thus,  cos  300°  ==  cos  (360°  —  60°)  =  cos  60°,  etc. 


I.   QUADRANT. 

II.   QUADRANT. 

III.   QUADRANT. 

IV.  QUADRANT. 

X  = 

<* 

180°  -a 

i&J'+a 

360°  -a 

or  -  a 

sin  x  = 

+  sin  a 

+  sin  a 

—  sin  a 

—  sin  a 

COS  J"  = 

+  cos  a 

—  cos  a 

—  cos  a 

-f  cos  a 

tan  x  = 

+  tan  a 

—  tan  a 

+  tan  a 

—  tan  a 

cot  x  = 

+  cot  a 

-  cot  a 

+  cot  a 

—  cot  a 

To  indicate  that  the  trigonometric  function  is  negative,  n  is  written 
after  its  logarithm. 


396  EXPLANATION    OF   THE    TABLES. 

40.  To  find  the  angle  corresponding  to  a  given  function,  find  the 
acute  angle  a  corresponding  thereto,  and  the  required  angle  will  be  a, 
i8o°±«,  or  360°  —  a,  according  to  the  quadrant  in  which  the  angle 
should  be  placed. 

41.  There  are  always  two  angles  less  than  360°  corresponding  to  any 
given  function.     Hence  there  will  be  ambiguity  in   the   result  unless 
some  condition  is  known  that  will  fix  the  angle  ;   thus,  if  the  sine  is 
positive,  the  angle  may  be  in  either  of  the  first  two  quadrants,  but  if  we 
also  know  that  the  cosine  is  negative,  the  angle  must  be  in  the  second 
quadrant. 

Given  One  Function  of  an  Angle,  to  find  Another  without  finding 
the  Angle. 

42.  Suppose  log  tan  «=  9.79361,  and  log  cos  «  is  sought.     On  page 
451  the  tabular  difference  for  log  tan  a  is  28,  and    that  for  log  cos  a 
is  8,  the  given  logarithmic  tangent  exceeding  9.79354  by  7.     Hence 
28:7  =  8:.*;    /.  x  =  2  =  correction    to    9.92905,    giving    log  cos  «  = 
9.92903. 

In  the  margin  are  tables  to  facilitate  the  process.  In  the  column 
headed  ^8g,  the  numerator  is  the  tabular  difference  for  the  logarithmic 
cosines,  and  the  denominator  that  for  the  logarithmic  tangents1.  The 
correction  for  the  logarithmic  cosine  will  be  o  when  the  given  logarith- 
mic tangent  exceeds  the  next  smaller  logarithmic  tangent,  found  in  the 
tables,  by  less  than  1.8,  i  for  an  excess  between  1.8  and  5.2,  5  for  an 
excess  between  15.8  and  19.2,  etc.  In  the  example  above,  the  excess 
was  7,  which  is  between  5.2  and  8.8,  so  that  the  correction  is  2. 

For  example,  if  we  have  given  the  logarithms  of  the  sides  of  a 
right-angled  triangle,  log  #  =  2. 98227  and  log  b  =  2.90255,  to  find  the 
hypotenuse,  we  use  the  formulas 

tan  «  =  2,  and;  =  -£-  =  -£-. 

b  sin «      cos  a 

The  value  of  log  tan  ft.  being  found  in 

log  a  =  2.98227  (i)         the  column  marked  L.  Tan  at  the  bot- 

.-.  log  sin  «  =9.88571  (4)         torn,  the  right  column  will  contain  the 

log£  =  2.90255  (2)         logarithmic   sine   of   the  corresponding 

.-.  log  tan  «  =  0.07972  (3)         angle.     Also,  the  correction  to  9.88563 

.-.    log<:  =  3.09656  (5)         is  20  x  \\,  which  we  find  to  be  8  from 

the  table  headed  £$. 

1  For  angles  <  45°. 


TABLE    XV. 


COMMON 


LOGARITHMS    OF    NUMBERS 


FROM     I     TO     IIOOO. 


Log. 


Log. 


N. 


Log, 


N, 


Log. 


N, 


Log. 


17 
18 

'9 
2O 


0.30  103 
0.47  712 

0.60  206 
0.69  897 
0.77815 

0.84510 
0.90  309 
0.95  424 


1.04139 
1.07  918 
1.11394 

.14613 
.17609 
.20412 

•23  045 

•25  S2? 
.27  875 


1.30  103 


2O 

21 
22 
23 

24 
25 
26 

27 
28 

29 

3O 


32 
33 

34 

35 

36 

37 
38 
39 

4O 


•3°  103 


1.32222 
1.34242 
I-36  173 
1.38021 
1-39794 
1.41  497 

1-43  136 
1.44716 
i  .46  240 


.49  136 


1.51851 

1.53  148 
1.54407 
I-55630 
1.56820 
I-57978 
1.59  106 


i. 60  206 


4O 


49 
50 

5' 
52 
53 
54 


i  .60  206 


6O 


1.61  278 
1.62325 
I-63347 

-64  345 
.65  321 
.66  276 

.67  210 

.68  124 

.69  020 


.69  897 


•70757 
.71  600 


•73  239 
•74036 
.74819 

•75  587 
•76  343 
-77085 


1-77815 


6O 

61 
62 
63 

64 


79 
8O 


[.77815 


I-78533 
1.79239 

J-79934 
1.80618 
1.81  291 

1.81  954 

1.82  607 
1.83251 
1.83885 


8O 

81 
82 
83 
84 


87 
88 

89 
90 


1.90309      IOO 


1.84510 


1.85  126 

1-85  733 
1.86332 

1.86923 
1.87506 

1. 88  08 1 

1.88649 
1.89209 

1.89  763 


•90309 


1.90849 
1.91  381 
1.91  908 

1.92428 
1.92942 
1.93450 

1-93952 
1.94448 
1-94939 


1.95424 


1.95904 
1.96379 
1.96848 

I.973I3 
1.97772 
1.98  227 

1.98677 
1.99123 
1.99564 


S'. 

6.46  373 
373 


T'. 
373 
373 


o°  o'  =  o" 
o   i  =  60 

O    2  =  120 


S".  T". 

4-68  557  557 

557  557 

557  557 


(397) 


398 


S'.  T'. 

N. 

L.    0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P,  P. 

366   385 

100 

ooooo 

043 

087 

130 

'73 

217 

260 

303 

346 

389 

AA       AQ       AO 

366  1  385 

101 

432 

475 

518 

561 

604 

647 

689 

732 

775 

817 

366 

366 

3^5 
386 

102 
103 

860 

oi  284 

945 
368 

988 
410 

"•030 
452 

*072 

494 

*SsI 

*i57 

578 

*i99 
620 

*242 
662 

2 

8.8    8.6   8.4 

u 

'366 
366 

386 
386 
386 

I04 

I05 
1  06 

703 

02  119 

745 
1  60 

572 

787 

202 

612 

828 
243 
653 

870 
284 
694 

912 

735 

953 
366 
776 

991 
407 
816 

*o36 

449 
857 

*o78 
49P 
898 

4  17.6  17.2  16.8 

5  22.O  21.5  2I.O 

6  26.425.825.2 

366 

387 

107 

938 

979 

*oi9 

*o6o 

*IOO 

*i4i 

*i8i 

*222 

*262 

*302 

730.830.1  29.4 

•  365 

387 

1  08 

03342 

383 

42.3 

463 

503 

S43 

583 

623 

66^ 

703 

835.234.433.6 

,365  387 

109 

743 

782 

822 

862 

902 

941 

981 

*O2I 

*o6o 

*IOO 

9!39-6  38.7  37-8 

365  387 

no 

04139 

179 

218 

258 

297 

336 

3/0 

415 

454 

493 

365  !  388 

in 

532 

57i 

610 

650 

689 

727 

766 

8oS 

844 

883 

365    388 

112 

922 

961 

999 

*07S 

*077 

*ii5 

•"192 

*27I 

*2faq 

1 

4.1  4.0  3.9 

365 

388 

"3 

05308 

346 

385 

423 

461 

538 

576 

614 

6S2 

2 

8.2    8.0    7.8 

365 
365 
364 

389 
339 
389 

114 

"5 

116 

690 
06070 
446 

483 

767 
145 
521 

805 
183 
558 

843 

221 

595 

881 
258 
633 

918 
296 
670 

956 
333 
7°7 

994 
371 
744 

408 
78l 

3  11.3  i^.u  11.7 
4  16.4  16.0  15.6 

5120.5  2O.O  19.5 

6  24.6  24.0  23.4 

364 

389 

117 

819 

856 

893 

930 

967 

*oo4 

*04i 

*o;8 

*ii5 

*I5I 

7  28.7  28.0  27.3 

364 

390 

118 

07188 

225- 

262 

298 

33  S 

372 

408 

44  S 

482 

51* 

8(32.832.031.2 

364   390 

119 

55J. 

591 

628 

664 

700 

737 

773 

809 

846 

882 

9:36.936.035.1 

364 

39° 

I2O 

918 

954 

990 

'171 

•'•243 

364 

391 

121 

08279 

3H 

3  So 

386 

422 

4S8 

493 

S29 

565 

600 

38     37     36 

!  363 

391 

122 

636 

672 

707 

743 

778 

814 

849 

884 

920 

955 

i 

3-»   3-7   3-o 

363 

391 

123 

991 

*026 

*o6i 

*o96 

*i67 

*202 

*237 

*272 

•307 

2 

7-6    7-4    7-2 

363 
363 
363 

391 

392 
392 

124 

125 
126 

09342 
691 
10037 

377 
726 
072 

412 
760 
106 

447 
795 
140 

482 
830 

209 

552 
243 

587 
934 
278 

621 
968 
312 

656 
346 

3 

4 

i 

11.4  1  1.  1  10.8 
15.2  14.8  14.4 
19.0  18.5  18.0 

363 
363 
362 

392 
393 
393 

I27 
128 
I29 

380 
721 
11059 

415 
755 
093 

449 
789 
126 

483 
823 
160 

857 
193 

551 
890 
227 

585 
924 
26l 

619 
958 
294 

653 

992 
327 

687 
*36? 

I 

9 

26.6  25.9  25.2 
30.4  29.6  28.8 

74-2  77.7   72.4 

362 

393 

ISO 

394 

428 

461 

494 

528 

561 

594 

628 

66  1 

694 

362 

S94 

131 

727 

760 

793 

826 

860 

893 

926 

9S9 

992 

*O24 

35     34     33 

362 

362 

394 
394 

132 
133 

12057 
385 

123 

45° 

156 

483 

189 
516 

222 
548 

$ 

287 
613 

646 

352 

678 

I 

3-5    3-4    3-3 
7.0   6.8    6.6 

362 
'361 

395 
395 

134 

710 
13033 

III 

775 
098 

808 

170 

840 
162 

872 

194 

226 

937 

2S8 

969 

290 

*OOI 

322 

3 

4 

10.5  10.2   9.9 
14.013.613.2 

395 

136 

354 

386 

418 

450 

481 

513 

545 

577 

609 

640 

5 

17.5  17.0  16.5 

'361     396 
361     396 
361     396 

137 
138 

139 

672 
988 
14301 

*7°4 
333 

*735 

767 
*082 

395 

799 
*u4 
426 

*83- 
457 

862 
*i76 
489 

520 

925 
*239 

551 

956 

*270 
582 

I 

9 

24.5  23.8  23.1 
28.0  27.2  26.4 

71.  C  7Q.6  2Q.7 

361 

360 

397 

140 

141 

613 

644 

if 

^06 

*OI4 

737 

768 

"•076 

799 
*io6 

829 

860 

89I 

32     31     30 

397 

922 

953 

*°45 

*i37 

*i68 

*I98 

360  |  397 

142 

15229 

259 

290 

320 

35' 

381 

412 

442 

473 

S03 

i 

3.2     3-1      3.0 

360 

3y8 

534 

564 

594 

625 

685 

715 

746 

776 

806 

2 

6.4    6.2    6.0 

36o 

)98 

144 

836 

866 

897 

927 

9S7 

987 

*oi7 

*047 

*io7 

3 

9.6  9.3  9.0 

360 

398 

16137 

167 

197 

227 

256 

286 

716 

346 

376 

406 

4 

12.8  12.4  12.0 

360 

399 

146 

435 

465 

495 

524 

554 

584 

613 

643 

673 

702 

5 

1  6.0  15.5  15.0 

359 

1359 
J359 

399 

399 

.100 

4s 
149 

ISO 

732 
17026 

761 

056 

348 

791 
085 
377 

820 
114 
406 
696 

850 
143 
435 

879 
'73 
464 

909 
202 
493 

938 
231 

-iff 

967 
260 

551 

997 
289 
580 

6  19.2  15.0  is.o 
7  22.4  21.7  21.0 
8:25.624.824.0 
9  28.8  27.9  27.0 

|  359    4°o 

609 

638 

667 

725 

754 

782 

840 

869 

N. 

L.    0 

1     |    2 

3 

4 

5678 

9 

P.  P. 

S.'      T.' 

S."      T." 

S."    T." 

i'    6.46373     373 

o°     i'=     60"  4-68557     557 

o°  19'=  1  140"    4-68557     558 

2             373      373 

o      2  =   120             557     557 

o    20=1200              557     558. 

10             373      373 

o      3  =   180             557     557 

o    21  =1260              557     558 

13             373     373 

o    16  =  960             557      558 

o    22-1320              557     558 

'4             372      373 

o    17=1020             557     558 

o    23=1380              557     558 

15             372      373 

o    18=1080             557     558 

o    24=1440              557     558 

o    19=1140             557      558 

o    25  =1500              557     558 

399 


!s/6. 

359 

359 
358 
358 
358 
358 
358 
358 
357 
357 
357 
357 
357 
356 

356 
356 
356 
356 
355 
355 

T'. 

46 

401 
4o 
401 

409 

402 
403 
4<>3 

404 

N. 

L.    0        1         23 

4 

5        6 

7 

8 

9 

P.P. 

ISO 

'51 

'53 

154 
|55 

157 
158 
159 
I6O 

161 
162 
163 
164 

169 
170 

171 
172 

173 

174 
'75 
176 

177 
178 
179 

180 

181 
182 
183 
184 
185 
1  86 
187 
1  88 
189 
190 
191 
192 
193 
194 

199 
200 

17609 
898 
18  184 
469 

752 
19033 
312 

590 
866 
20  140 
412" 

638 

667 

696 

725 

754 

*04I 

327 
611 

893 
173 
45  ! 

*728 
276 

782 

81 

840 

869 

i 

2 

3 

4 

7 

s 

9 
i 

2 

3 
4 
5 
6 

8 
9 

i 

2 

3 

4 

1 

9 
i 

2 

3 
4 

1 

7 
8 
9 

2S 

2 

II 
14 
17 
20 

23 

26 

27 

2 

IO 
13 

16 
18 

21 

24 

2 

3 

4 
5 

6 

i 

24 

2 

4 
7 
9 

12 
14 

1  6 
19 

21 

22 

2 

4 
6 
8 
ii 
13 

'5 

17 
19 

28 

9      2.8 
8      5.6 
7     8.4 

6      II.  2 

5    HO 
4    16.8 
3    19.6 

2     22.4 
I      25.2 

26 

7      2.6 
4      5-2 
i      7.8 
8    10.4 

5    13-° 
2    15.6 
9    1  8.2 
6    20.8 
3    23.4 

25 

2-5 

5-o 

7-5 

IO.O 

12.5 
15.0 
17-5 

20.0 
22-5 

23 

4      2.3 
8     4.6 
2      6.9 
6      9.2 
o    11.5 
4    13-8 
8    16.1 
2    18.4 
6    20.7 

21 

2        2.1 

4      4-2 
6      6.3 
8      8.4 
o    10.5 

2     12.6 

4    H-7 
6    1  6.8 
8    18.9 

926 
213 
498 
780 
06  1 
340 
618 
893 
167 
439 

955 
241 
526 
808 
089 
368 

645 
921 
194 

984 
270 

554 

837 
117 

396 

673 
948 

222 

"013 
298 
583 
865 

H5 
424 

700 
976 
249 

•"070 
355 
639 
921 
20  1 
479 

303 

384 
667 

949 

229 

5°7 

*o88 
330 
602 

-127 
412 
696 

977 

257 
535 
811 
*o85 
358 

"156 
441 
724 
*oo5 

I5 
562 

838 

*II2 

385 

404 

466 

493 

520 

548 

575 

629 

656 
*I92 

458 

722 

246 
505 

J63 

404 

405 
405 
4u6 
406 
406 
407 
407 
408 

408 

683 
952 

21  219 
484 
748 
22  01  1 

272 
531 
789 

710 
978 
245 
511 
775 
037 
298 

si 

737 

*005 

272 

537 
80  1 
063 
324 

583 
840 

763 

*032 

299 

564 
827 

089 

350 
608 

866 

*o9° 
325 
590 
854 
"5 
376 
634 
891 

*o87 
352 
617 
880 
141 
401 
660 
917 

844 

*II2 

378 

643 
906 
167 

427 
686 
943 

871 
*i39 

405 
669 
932 
194 

453 
712 
968 

898 
*i65 

696 
958 

220 

479 
737 
994 

355 

23045 

070 

096 

350 
603 

855 
105 
353 
601 

846 
091 

334 
575 

121 

M7 

172 

198 

223 

249 

274 

355 
354 
354 
354 
354 
354 
353 
353 
353 

408 
409 

4<>y 

410 
41 

41 

300 

553 
805 

24055 
304 

797 

25042 
285 

325 
578 
830 

080 
329 
576 
822 
066 
310 

376 
629 
880 
I30 

378 
625 

87I 
115 

3S8 

401 
654 
905 
155 
403 
650 

895 
139 
382 
624 

426 
679 
930 
1  80 
428 
674 
920 
164 
406 

452 
704 

955 
204 
452 
699 

944 

1  88 

477 
729 
980 

229 
477 
724 
969 

212 

455 

502 
754 
*oo3 

254 
502 
748 
993 
237 
479 

528 
779 
•"030 

279 
527 
773 
*oi8 
261 
J03 
744 

353 

41 

527- 
768 
26007 
245 
482 
717 
95' 
27184 
416 
646 

600 

648 

672 

696 

720 

353 

352 
352 

352 

S 

!35° 
35° 
350 
350 
350 
349 
349 
349 
349 
348 

41 
41 

41 

4i 
41 

41 

792 

031 
269 

505 

975 
207 
439 
669 

816 
055 
293 

529 
764 
998 
231 
462 
692 

840 
079 

ii 

*02I 

254 

485 
715 

864 

102 
340 
576 

811 

277 
508 
738 

888 
126 
364 
600 
834 
*o68 

300 

761 

912 
150 
387 
623 

858 

*09I 

323 

554 
784 

935 
411 

647 
88  1 
*U4 
346 
577 
Jo7 

959 
198 

435 
670 
^905 

370 
600 
830 

983 

221 

458 
694 
928 

*i6i 

393 
623 
852 
*o8i 

41 

418 
418 
419 
419 
420 
420 
421 
421 

875 
28  103 
330 
SS6 
780 
29003 
226 

447 
667 
885 

898 

921 

944 

623 

847 
070 
292 

513 

732 

967 

989 

*OI2 

126 

803 
026 
248 

469 
688 
907 

149 

375 
601 

825 
048 
270 
491 
710 
929 

194 
421 
646 
870 
092 
3H 
535 
754 
973 

217 

443 
668 

892 

"5 

336 

557 
776 
994 

240 
466 
69I 
914 

'37 

358 
579 
798 
*oi6 

262 
488 
7'3 
937 

601 
820 
•"038 

285 
5" 
735 
959 
181 
403 
623 

*o6o 

307 
533 
758 
981 
203 
425 

645 
863 
*o8i 

348 

439 

30103 

'23 

146 

1  68 

190 

211 

233 

255 

276 

298 

N. 

L.    0 

1 

2    i     3 

4 

5678 

9    |          P.P. 

S.'      T.' 
i'    6.46  373      373 

2             373      373 

o°    2'  = 
o      3  = 

S."      T." 
120"   4-68557      557 
'80             557      557 
240             557      SS8 

S."     T." 
o°  28'=  1680"    4-68557    558 
o    29=1740              557     559 
o    30=1800              557     559 

15             372      373 
20             372      373 

0         .} 

o    25  =1500             557     558 
o    26=1560             557      558 
o    27=1620             557     558 
o    28=1680             557     558 

o    31  -1860              557     559 
o    32  =1920              557     559 
o    33  =1980  .           557     559 
o    34=2040              557     550 

400 


N. 

L.  0 

I 

2 

3 

4 

5    6 

7 

8 

9 

P.  P. 

200 

20  1 

202 
203 
204 
205 
206 

208 
209 

210 

211 
212 
213 
214 
215 

216 

2I7 

218 

219 

220 

221 
222 
223 
224 
225 
226 
227 
228 
229 

230 

231 
232 
233 
234 
235 
236 

237 
238 
239 
24O 

241 
242 
243 
244 

245 
246 

247 
248 
249 

250 

30103 

125 

146 

1  68 

190 

211 

233 

253 

8| 

899 

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323 

534 
744 
952 
160 

276 

298 

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2 

3 
4 
5  i 
6  i 

7  i 
8  i 

9  i 

i 

2 

3 
4 
5 
6 

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9 
i 

2 

3 

4 

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1 

9 
i 

2 

3 
4 

I 

7 
8 

9 
i 

2 

3 
4 

I 

9 

22   21 

2.2   2.1 
4-4   4-2 

6.6  6.3 
8.8  8.4 
i.o  10.5 
3.2  12.6 
5-4  14-7 
7.6  16.8 
9.8  18.9 

20 

2.0 
4.0 

6.0 
8.0 

1  0.0 
12.0 
14.0 

1  6.0 
1  8.0 

19 

i-9 

3-8 

\l 

9-5 
11.4 

13-3 
J5-2 
17.1 

18 

1.8 
3-6 
5-4 
7-2 
9-o 
10.8 

12.6 

14.4 
16.2 

17 

i-7 

3-4 

8 

8.5 

10.2 
II.9 
I3.6 
15-3 

320 
535 
75° 
963 
3i  175 
387 

597 
806 
32015 

341 
557 
771 

984 
197 
408 
618 
827 
035 

363 
57« 
792 
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218 
429 

639 
848 
056 

3«4 
600 
814 

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239 
45° 
660 
869 
077 

406 
621 

835 
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260 
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681 
890 
098 

428 

643 

856 

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281 
492 
702 
911 
118 

449 
664 
878 
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302 
513 
723 
93i 
139 

492 
707 
920 

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343 
555 
763 
973 
181 

5»4 

728 
942 

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366 
576 
785 
994 

201 

222 
^28" 
634 
838 

33041 
244 

445 
646 
846 
34044 

243 

263 

284 

305 

325 

346 

366 

387 

408 

449 
654 

858 

062 
264 
465 
666 
866 
064 

469 
675 
879 

082 
284 
486 
686 
885 
084 

490 
695 
899 
I  O2 
304 
506 

706 

905 
104 

510 
715 
919 

122 

325 
526 
726 
925 
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53I 
736 

940 
H3 

34I 
546 

746 
945 
143 

552 
756 
960 

163 
365 
566 
766 
965 
163 

572 
777 
980 

183 

385 
586 
786 
983 
183 

593 
797 

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203 
405 
606 

806 
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203 

613 

818 

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224 

425 
626 

826 

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223 

242 
439 
635 
830 

35°23 
218 
411 
603 
793 
984 

36i73 

262 

282 

3OI 

321 

34i 

36i 

380 

400 

420 

459 
655 
850 

044 
238 
43° 
622 

813 
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479 
674 
869 
064 
257 
449 
641 
832 

*02I 

498 
694 
889 

083 
276 
468 
660 

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"•040 

518 

7'3 

908 

102 

295 
488 

679 
870 

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537 
733 

928 

122 
315 

507 

698 
889 

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557 
753 
947 
141 
334 
526 

717 
908 
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577 
772 

967 
160 
353 
545 
736 
927 
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596 
792 
986 
180 
372 
564 

755 
946 

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616 
811 

-005 

199 

774 

»965 
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192 

211 

229 

248 

267 

286 

303 

324 

342 

361 
549 
736 
922 

37I07 
291 

475 
658 
840 

3802T 

380 
568 
754 
940 
125 
310 

493 
676 
858 

399 
586 
773 
959 

% 
% 

876 

418 
605 
791 

977 
162 
346 
S3o 
712 
894 

436 
624 
810 
996 
181 
365 
548 
73i 
912 

455 
642 
829 
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199 
383 
566 
749 
93i 

474 
66  1 
847 

*o33 
218 
401 

583 
767 
949 

493 
680 
866 

*05I 

236 
420 
603 
785 
967 

511 

698 
884 
"070 
254 
438 
621 
803 
985 
1  66 

530 
717 
903 

*o88 
273 
457 

639 
822 
*oo3 

039 

°57 

°75 

093 

112 

130 

148 

184 

202 
382 
56i 

739 

917 

39094 
270 

445 
620 

794 

220 
399 
578 
757 
934 
in 

287 
463 
637 

238 
417 
596 

775 
952 
129 

305 
480 

655 

256 

435 
614 

792 
970 
146 
322 
498 
672 

274 
453 
632 
810 
987 
164 

340 
5'5 
690 

292 

471 
650 

828 

*oo5 
182 

358 
533 
707 

310 

489 
668 

846 

*023 

199 

375 
550 
724 

328 
5°7 
686 

863 

*04I 

217 

393 
568 
742 

346 
523 
703 
88  1 
"•058 
233 
410 
585 
759 

364 
543 
721 

899 
"•076 
252 
428 
602 
777 

811 

829 

846 

863 

881 

898 

915 

933 

95° 

N. 

L.  0    1 

234 

5 

6 

7 

8 

9 

P.P. 

S.'   T.' 
2'  6.46373  373 
3     373  373 

0° 
0 

S."  T." 
3'=  180"  4-68557  557 
4=  240     557  558 

5  =  3oo      557  558 

0°  . 

o  . 
o 

S."  T." 
56'  =  2160"  4.68  557  559 
57  =  2220     557  559 
58  =  2280     557  559 
59  =  2340     557  559 
jo  =  2400     557  559 
\i  =  2460     556  560 
12  =  2520     556  560 

20     372  373 
25     372  373 

o 

o  33  =  1980      557  559 
o  34  =  2040      557  559 
o  35  =  2100      557  559 
o  36  =  2160      557  559 

o 

o  < 
o  < 
o 

401 


N. 

L.  0 

1 

234 

5 

6 

7 

8 

9 

P.P. 

25O 

251 
252 

253 
254 
255 
256 

257 
258 
259 
26O 

261 
262 
263 

264 

265 
266 
267 
268 
269 

270 

271 
272 
273 
274 
275 
276 

277 
278 
279 

280 

281 
282 
283 
284 
285 
286 
287 
288 
289 

29O 

291 
292 
293 

294 
295 
296 
297 
298 
299 

300 

39794 

811 

829 

846 

863 

88  1 

898 

9'5 

933 

95° 

i 

2 

3 
4 

I 

9 

2 

3 
4 

5 
6 

I 

9 

i 

2 

3 

4 

I 

7 
8 
9 

i 

2 

3 
4 
5 
6 

1 

9 

2 

3 

4 

9 

18 

1.8 
3-6 
5-4 
7-2 
9.0 
10  8 

12.6 

14.4 
16.2 

17 

i-7 
3-4 

8 

8-5 

10.2 
1  1.9 
13-6 

»S-3 

16 

1.6 

it 

6.4 

8.0 
9.6 

II.  2 

12.8 

14.4 

15 

i-5 
3-o 

tl 

7-5  • 
9.0 
10.5 

12.0 

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14 
1.4 

2.8 

4.2 
5-6 
7.0 
8.4 
9.8 

II.  2 
12.6 

967 
40  140 
312 

483 
654 
824 

993 
41  162 
330_ 
497 
664 
830 
996 
42  160 

325 

488 

651 
813 
975 

985 
157 
329 
500 
671 
841 

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179 
347 

*002 
175 
346 
5^8 

688 
858 

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196 
363 

*oi9 
192 
364 
535 
705 
875 
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212 
380 

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209 
38i 

552 
722 
892 
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229 
397 

*°54 
226 

398 
569 
739 
909 

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246 
414 

*07I 

243 
415 
586 
756 
926 

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263 

430 

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261 
432 
603 
773 
943 
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280 
447 

*io6 
278 
449 
620 
790 
960 

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296 
464 

"123 

295 
466 

637 
807 
976 

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3i3 
481 

647 

5'4 

531 

547 

504 

58i 

597 

614 

631 

68  1 
847 

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177 
341 
504 
667 
830 
991 

697 

863 

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193 

357 
521 
684 
846 
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714 

880 
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210 

374 
537 
700 
862 

*024. 

& 

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226 
390 

553 
716 
878 
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747 
913 
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243 
406 

57° 
732 
894 
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764 
929 
"095 
259 
423 
586 

749 
911 

"•072 

780 
946 
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275 
439 
602 

765 
927 
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797 
963 

*I27 

292 

455 
619 

781 

943 
"•104 

814 
979 
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308 
472 
63? 
797 
959 

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43  136 

152 

169 
329 
489 
648 

807 
965 

122 

279 
436 

592 

~34T 
505 
664 

823 
981 
138 
295 

45  l 
607 

201 

217 

233 

249 

265 

425 
584 
743 
902 
"059 
217 

373 
529 
685 

28l 

297 
457 
616 

775 
933 
44091 

248 
404 
560 

3U 

473 
632 

791 

949 
107 

264 
420 
576 

361 

521 
680 

838 
996 
154 

3" 

467 
623 

377 

ig 

854 

*OI2 
170 
326 
483 
638 

393 
553 
712 

870 

*028 

185 
342 
498 
654 

409 

569 
727 

886 
*o44 
20  1 
358 
5'4 
669 

441 
600 

759 
917 
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232 

389 
545 
700 

716 

73i 

_747_ 
902 
056 
209 
362 

® 

818 
969 
1  20 

762 

778 

793 

809 
963 

117 

271 

423 
576 
728 
879 

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1  80 

824 

840 

855 

871 
45025 
179 

332 
484 
637 

788 

939 
46090 

886 
040 
194 

347 
500 
652 
803 
954 
105 

917 
071 
22? 

378 
53° 
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834 
984 

135 

932 
086 
240 

393 
545 
697 

849 

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150 

948 

102 

255 
408 
561 
7I2 

864 

"015 
165 

979 
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286 

439 
59i 
743 
894 
*°45 
'95 

994 
148 
301 

454 
606 
758 
909 
*o6o 

210 

*OIO 

163 
317 
469 
621 

773 
924 
*°75 
225 

240 

255 

270 

285 

300 

315 

330 

3.45 

359 

374 

389 
538 
687 

835 

982 

47  I29 
276 
422 
j>67_ 
712 

404 

553 
702 

830 

997 

144 

290 
436 
582 

419 

568 
716 
864 

*OI2 
159 
305 
451 
596 

434 
583 
731 
879 

*026 

173 
319 

465 

611 

449 
598 
746 
894 

*04I 

1  88 

334 
480 
625 

464 
613 
761 
909 
•"056 
202 

349 
494 
640 

479 
627 
776 

923 

*O70 
217 

363 
5°9 
654 

494 
642 
790 

938 
"085 
232 

378 
524 
669 

509 

657 
805 

953 

*IOO 

246 
392 

538 
683 

523 
672 
820 
967 
*M4 
261 
407 

13 

727 

74I 

756 

77° 

784 

799 

813 

828 

842 

N, 

L.  0 

1234 

5 

6    7 

8 

9 

P.  P. 

S.'  T.' 

2'  6.46373  373 
3     373  373 

S."  T." 
o°  4'=  240"  4.68  557  558 
o  5  =  3°o      557  558 

0° 

o 

S."  T." 
tS'=-27oo»  4-68556  560 
$6  =  2760      556  560 
17  =  2820     556  560 
j8  =  2880      556  560 
W  =  2940      556  560 
jo  =  3000      556  561 

25      372  373 
26     372  373 
27     372  374 
30     372  374 

o  41  =  2460      556  560 
o  42  =  2520      556  560 
o  43  =  2580      556  560 
o  44  =  2640      556  560 
o  45  =  2700      556  560 

0 
0 
0 

o 

R'M'D  SURV.  —  26 


402 


N. 

L.  0 

1 

234 

6 

7 

9 

P.P. 

300 

301 
302 
303 
304 
305 
306 

307 
308 
309 
310 

3" 
312 

313 
3H 
315 
316 

317 
3i8 
319 
32O 

321 
322 
323 
324 

9 

% 

329 
330 

33i 
332 
333 
334 
335 
336 

337 
338 
339 
340 

34i 
342 
343 
344 
345 
346 

347 
348 
349 
350 

47712 

727 

741 

756 

770 

914 
058 

202 

344 
487 
629 
770 
911 

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784 

799 

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828 

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i 

2 

3 
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9 
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2 

3 
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2 

3 
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7 
8 

9 
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2 

3 
4 
5 
6 

9 

15 

!-5 

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4-5 
6.0 

7-5 
9.0 
10.5 

12.0 

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14 

1-4 

2.8 

4.2 
5-6 
7.0 
8.4 
9.8 

II.  2 
12.6 

13 

i-3 

2.6 

3-9 

6:52 
7.8 

9-i 
10.4 
11.7 

12 

1.2 

2.4 
3-6 
4.8 

6.0 
7.2 
8.4 
9.6 
10.8 

857 
48001 
144 
287 
430 
572 
7J4 
855 
996 

871 
015 
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302 
444 
586 
728 
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885 
029 
173 
316 
458 
601 
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883 

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900 
044 
187 
330 
473 
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756 
897 
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929 

073 
216 

359 
501 
643 

783 
920 
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943 
087 
230 

373 
5'5 
657 
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958 

101 

244 

387 
530 
671 

813 
954 
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234 

972 

116 

259 
401 

6-81 

827 
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986 
130 
273 
416 
558 
700 

841 
982 

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49136 

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164 

178 

192 

206 

220 

248 

262 

276 
415 
554 
693 
831 
969 

50  106 
243 

379 

290 
429 
568 
707 

845 
982 

120 

256 

393 

304 

443 
582 

721 
859 
996 

133 
270 
406 

3i8 

457 
596 

734 
872 

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284 
420 

332 

471 
610 

748 
886 

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161 

297 
433 

346 
485 
624 
762 
900 
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174 
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447 

360 

499 
638 

776 
914 
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188 

323 
461 

374 
5'3 
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790 
927 
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202 
338 

474 

388 

SI 

803 
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215 

$ 

402 

679 
817 

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229 
365 
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5^5 

529 

542 

556 

569 

583 

596 

610 

623 

637 

651 
786 
920 

5'°53 
1  88 
322 

455 
587 
720 

85J_ 
983 
52114 

244 
375 
5°4 
634 

763 
892 
53020 

664 
799 
934 
068 

202 

335 
468 
601 
733 

678 
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947 
081 
215 
348 
481 
614 
746 

691 
826 
961 

093 
228 
362 

495 
627 

759 

703 
840 
974 
108 
242 
375 
508 
640 
772 

718 
853 
987 

121 

IS 

521 
654 
786 

732 
866 

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133 
268 
402 

534 
667 
799 

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148 
282 
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548 
680 
812 

759 
893 

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162 

295 
428 

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693 

825 

772 
907 

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175 
308 
441 

574 
706 
838 

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878 

891 

904 

917 

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957 

970 

996 
127 

257 
388 

517 
647 

776 
905 
033 

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140 
270 
401 

530 
660 

789 
917 
046 

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284 

414 
543 
673 
802 
930 
058 

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1  66 
297 

427 
556 
686 

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943 
071 

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179 
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440 
569 
699 
827 

956 
084 

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192 
323 
453 
582 
711 
840 
969 
097 

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205 
336 
466 
595 
724 

853 
982 
no 

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218 
349 

479 
608 
737 
866 
994 

122 

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$ 

492 
621 

75° 
879 

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i35 

148 

161 

173 

186 

199 

212 

224 

237 

230    263 

275 
403 
529 
656 
782 
908 

54033 
158 
283 

288 
415 
542 
668 
794 
920 

045 
170 
295 

301 
428 
555 
68  1 
807 
933 
058 
183 
307 

3H 
441 

567 
694 
820 
945 
070 

*95 
320 

326 

453 
580 

706 
832 
958 
083 
208 
332 

339 
466 

593 
719 
845 
970 

095 

220 

343 

352 
479 
605 

732 
857 
983 
108 
233 
357 

364 
491 
618 

744 
870 

995 
1  20 
245 
370 

377 
5°4 
631 

757 
882 
*oo8 

133 
258 
382 

390 
5i7 

643 
769 
895 

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'45 
270 

394 

407 

419 

432 

444 

456 

469 

481 

494 

506 

5i8 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.  P. 

S.'   T.' 
3'  646373  373 

4     373  373 

S."  T." 
o°  5'=  300"  4-68557  558 

o  6  =  360      557  558 

o° 
o 

S."  T." 
54'  =3240"  4-68556  561 
55  =  3300     556  561 
56  =  3360     556  561 
57  =  3420     555  561 
58  =  3480     555  562 
59  =  3540     555  562 

30     372  374 
35     372  374 

o  50  =  3000      556  561 
051=  3060      556  561 
o  52  =  3120      556  561 
o  53  =  3180      556  561 
o  54  =  3240      556  56' 

o 

o 

0 
0 

403 


N.  |  L.  0  |   1 

2 

3 

4 

5    6    7  i  8 

9 

P.P. 

35O 

351 

352 
353 
354 
355 
356 

357 
358 
359 
36O 

36i 
362 

363 
364 

3^I 
366 

367 
368 

369 
370 

37i 
372 
373 
374 
375 
376 

377 
378 
379 
38O 

38i 
382 
383 
384 
385 
386 

387 
388 
389 
390 

39i 
392 
393 
394 
395 
396 

397 
398 
399 
400 

54407   419 

432 

444 

456 

469 

481   494   506 

5'8 

i 

2 

3 
4 

1 
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9 
i 

2 

3 
4 
5 
6 

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9 

i 

2 

3 
4 

9 

2 

3 
4 

7 

8 

9 

13 
1.3 

2.6 

3-9 

£ 

7-8 
9.1 
10.4 
u.7 

12 

1.2 

2.4 

3-6 
4.8 
6.0 

7-2 

8.4 
9.6 

10.8 
11 

2.2 

3-3 
4-4 
5-5 
6.6 

88 

O.O 

9-9 
10 

I.O 

2.0 

3-o 

4.0 

5-o 
6.0 
7-o 
8.0 
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P.  P. 

S.'  T.' 
3'  6.46  373  373 
4     373  373 

0° 
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S."  T." 
5'=  300"  4.68  557  558 
6=  360      557  558 

7  =  420      557  558 

« 

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i'=366o"  4-68555  562 
2  =  3720      555  562 
3  =  378o      555  562 
4  =  3840      555  563 
5  =  3900      555  563 
6  =  3960      555  563 
7  =  4020      555  563 

35      372  374 
39     372  374 
40     372  375 

0 

o  58  =  3480      555  562 
o  59  =  3540      555  562 
i  o  =  3600     555  562 
i  i  =  3660     555  562 

404 


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1 

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3 

4 

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P.  P. 

400 

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402 
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409 

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411 
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417 
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421 
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423 
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365 

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414 

424 

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640 

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464 
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273 

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360 

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1 

2    3 

4 

5 

6    7 

8 

9 

P.P. 

S.'  T.' 
4'  6.46373  373 

5     373  373 

S."  T." 
o°  6'=  360"  4.68  557  558 
o  7  =  420      557  558 
o  8=  48o      557  558 

0  9'=  41 
10  =  42 
ii  =  42 

S."  T." 

40"  4-68555  563 
oo      554  563 
60      554  564 
20      554  564 
80      554  564 
40      554  564 
oo      554  564 

40     372  375 
42     372  375 
43     37i  375 
44     371  375 
45     37i  375 

i  6  =  3960      555  563 
i  7=4020      555  563 
i  8  =  4080     555  563 
i  9  =  4140      555  563 

12-43 

13  =43 
14  =44 

15=45 

405 


N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

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P.P. 

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452 
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455 
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457 
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466 

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474 
475 
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L.  0 

1   |  2    3    4 

5  |  6    7    8    9 

P.  P. 

S.'   T.' 
4'  6.46  373  373 
5      373   373 

0° 

o 

S."   T." 
7'=  420"  4-68557  558 

8  =  480     557  558 
9  =  540     557  558 

0  i 

i 

8'=46 
9=47 

S."  T." 
80"  4-68554  565 
40      554  565 
00      554  565 
60      553  566 
20      553  566 
80      553  566 
40      553  566 

45     371   375 
48     371   375 
49      37'   376 
50      37'   376 

o  =48 
i  =48 
2=49 

3  =49 
14=50 

i  15  =4500     554  564 
i  16  =4560     554  565 
i  17  =4620     554  565 
i  18=4680     554  565 

2 

406 


N. 

L.  0 

I 

2    3  |  4 

5  j  6 

7    09 

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738 
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981 

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143 
223 

583 
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231 
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247   255 

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280 

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296 

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360 
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5 

6 

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8 

9 

P,  P. 

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5'  646  373  373 
6     373  373 

0° 
0 

S."  T." 
8'=  480"  4-68557  558 
9=  540      557  558 
o=  600      557  558 

0  26'=  5 
27  =  5 

S."  T." 
160"  4-68  553  567 
220     553  567 
280     553  567 
340     553  567 
400     553  567 
460     552  568 
520     552  568 

5°     371  376 
55     37i  376 

0  I 

28-5 
29  =  5 
30  =  5 
3i  =5 
32  =  5 

i  23  =  4980      553  566 
i  24  =  5040      553  566 
•i  25  =  5100      553  566 
i  26  =  5160      553  567 

407 


1  N* 

L.  0    1   1  2  |  3    4    5    6    7    8 

9 

P.P. 

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551 
552 
553 
554 
555 
556 

557 
558 
559 
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56i 
562 
563 
564 

566 

567 
568 
569 
570 

57i 
572 
573 
574 
575 
576 

577 
578 
579 
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£ 

583 
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585 
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587 
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589 
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592 
593 
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595 
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598 
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359 
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593 
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288 

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523 
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539 
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547 
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241 
320 

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476 
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632 
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249 
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406 
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562 
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718 
796 

178 
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570 
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726 
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358 
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664 
740 
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89i 
967 
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118 

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268 

904 
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136 
213 
289 
366 
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519 

912 

989 
066 

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220 
297 

374 
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526 

920 
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074 

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228 

305 
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458 
534 

927 
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082 

159 
236 
312 

389 
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542 

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320 

397 
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549 

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097 

174 
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105 
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504 
580 

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679 
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133 

208 
283 

610 

618 

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633 

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671 
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823 
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050 

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200 

275 

686 
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140 

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290 

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770 
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921 

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072 

148 
223 
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230 

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785 
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087 

163 
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103 

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350 

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567 

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574 
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433 
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967 
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701 

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173 
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539 
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327 
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1  88 
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335 
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634 
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203 
276 
349 
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495 
568 

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283 

357 
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503 
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7.21 

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217 
291 
364 
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583 
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728 
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223 
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L.  0 

1    2    3 

4 

56789 

P.P. 

S.'  T.' 

6'  6.46  373  373 

0° 

S."  T." 
)'=  540"  4-68557  558 

D  =  600      557  558 

°  3 

2 

5'=  57 

S."  T." 
oo"  4.68  552  569 
60      552  569 

20        552  569 
80        552  569 
40        551  569 

oo      551  570 

55      371  376 
56     371  376 
57     37i  377 
58     371  377 
59     37°  377 
60     37°  377 

O  I 

6-57 

7  =  58 
8  =  58 

9  =  59 

o  =  60 

31  =  5460    552  568 
32  =  5520    552  568 

33  =  558o      SS2  568 
34  =  5640      552  568 
35  =  57°°      552  569 

3 
3 
3 
4 

408 


N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

p.p. 

600 

601 
602 
603 
604 
605 
606 
607 
608 
609 
610 
6u 
612 
613 
614 

615 
616 

617 
618 
619 
620 

621 
622 
623 
624 
625 
626 
627 
628 
629 

630 

63. 
632 
633 
634 
635 
636 

637 
638 

639 
640 

641 
642 
643 
644 

645 
646 

647 
648 
649 
650 

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822 

830 

837 

844 

851 

859 

866 

873 

880 

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2 

3 
4 

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7 
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9 

2 

3 

4 

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9 

3 

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7 
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2.8 

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4-9 
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6 

0.6 

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5-4 

887 
960 
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176 
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319 
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533 
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169 

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118 
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283 

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540 

547 

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106 
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618 

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183 

625 
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155 
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668 
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092 
162 
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239 

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253 

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295 

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309 
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323 
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532 
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2 

3 

4 

5 

6 

789 

P.  P. 

S.'   T.' 
6  6.46  373   373 

7      373   373 

S."  T." 
o°  io'=  600"  4.68  557  558 

o  ii  —  660     557  558 

0  44'  =  62 
45  =63 

S."  T." 
40"  4.68  551  571 
oo      551  57i 
60      551  571 

20         550   572 

80    550  572 
40    550  572 

60     37°  377 
63     37°  377 
64     370  378 

65     370  378 

40  =6000     551   570 
41  =6060     551   570 
42  =6120     551   570 
43  =6180      551   570 
44  =6240      551   571 

46-63 

47  =64 
48=64 

49  =65 

409 


N. 

L.  0 

1 

2 

3 

4 

5 

6 

7    8 

9 

P 

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650 

651 

652 

653 
654 

657 
658 
659 
660 

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667 
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671 
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673 
674 
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677 
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692 
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432 
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334 

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145 

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270 

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512 
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286 

348 
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167 
230 
292 

354 
417 
479 

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173 
236 
298 
361 
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053 
117 

1  80 
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305 
367 
429 
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373 
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067 
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522 

528 

535 

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547   553 

559 

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N, 

L.  0 

1    2    3  |  4 

5 

6789 

P,  P. 

S.'   T.' 
6'  6.46  373  373 

7      373  373 

0° 

o 

S."  T." 
io'=  600"  4.68  557  558 
ii  =  660      557  558 
12=  720      557  558 

°  5 

i 

i 

S."  T." 
i'  =6660"  4-68530  573 
2  =  6720:     530  573 
3  =  6780      530  573 
4  =  6840      530  573 
5  =  6900      549  574 
6  =  6960      549  574 
7  =  7020      549  574 

65      37°  378 
69     37°  378 
7°     37°  379 

0 

I  48  =  6480    550  572 

i  49  =  6540      550  .572 
i  50  =  6600      550  572 
i  51  =6660      530  573 

c 

410 


N. 

L.  0 

1 

2 

3    4 

5    6 

7    8 

9 

P.  P. 

7OO 

701 
702 
703 
704 
705 
706 
707 
708 
709 

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711 
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720 

721 
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634 
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714 
763 
812 
861 
910 

522 

57i 
621 

670 
719 
768 
817 
866 
9i5 

527 

5A 
626 

67? 

724 

773 
822 
871 
919 

532 

!81 
630 

680 
729 

778 

827 
876 
924 

537 
586 

635 
683 
734 
783 
832 
880 
929 

939 
988 
95036 
085 

231 
279 
328 
376 

944 

949 

954 

959 
*oo7 
056 
103 
153 

202 
250 
299 

347 
395 

963 

968 

973 

978 

983 

993 
041 
090 

236 
284 
332 
38i 

998 
046 

095 

143 
192 
240 

289 

l& 

*OO2 

OS1 

100 

148 

197 
245 
294 
342 
390 

*OI2 
06  1 
109 

158 
207 

255 
303 
352 
400 

*oi7 
066 
114 
163 

211 

260 
308 

357 
405 

*O22 
071 
II9 

1  68 
216 
263 

3*3 
361 
410 

*O27 

075 
124 

173 

221 
270 

318 
366 
4f| 

*032 

080 
129 

177 

226 
274 
323 

371 
419 

424 

429 

434 

439 

444 

448 

453 

458 

463 

468 

N. 

L.  0 

1   |  2 

3  |  4 

5    6 

7    8 

9 

P.  P; 

S.'   T.' 

8'  6.46373  373 
9      373   373 

S."  T." 
o°  14'=  840"  4.68  557  558 

o  15  =  900     557  558 

2° 

2 

S."  T." 
25'  =  87oo"  4-68543  583 
26  =8760       544  584 
27  =8820       544  584 
28  =8880       544  584 
29  =8940       544  583 
30  =9000      544  585 

85     368  381 
86      368  382 
89      368  382 
90      368  383 

2  21  =8460     545   582 
2  22  =8520     545   582 
2  23  =8580     54?  583 
2  24  =8640     543  583 
2  25  =8700     543  583 

2 
2 
2 

414 


N. 

L.  0 

1 

3  |  4    5    6 

7    8 

9 

P.P. 

900 

901 
902 
903 
904 

9°5 
906 

907 
908 
909 

9IO 

911 
912 
913 
914 

915 
916 

917 
918 
919 

920 

921 
922 
923 
924 
925 
926 

927 
"928 
929 
93O 

93i 
932 
933 
934 
935 
936 

937 
938 
939 
940 

941 
942 
943 
944 
945 
946 

947 
948 
949 
950 

95424 
472 

5f 
569 

617 
665 
713 
761 
809 
856 

429 

434 

439 

444 

448 

501 
53o 
598 
646 
694 
742 

789 

Si 

458 

463 

468 

i 

2 

3 
4 

I 

9 
i 

2 

3 

4 

I 

7 
8 
9 

5 

o-5 

I  JO 

i-5 

2.0 

2-5 
3-o 

3-5 
4.0 

4-5 

4 

0.4 
0.8 

1.2 

1.6 

2.0 

2-4 
2.8 
3-2 

3-6 

477 
525 
574 
622 
670 
718 
766 

813 
861 

482 
530 
578 
626 
674 
722 

77° 
818 
866 

487 

535 
583 

631 
679 
727 

775 
823 
871 

492 
540 
588 
636 
684 
732 
780 
828 
875 

497 
543 
593 
641 
689 
737 

783 
832 
880 

506 

554 
602 

650 
698 
746 

794 
842 
890 

5" 
559 
607 

655 
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75  1 
799 
847 
893 

516 

564 

612 

660 
708 
756 
804 
852 
899 

904 

909 

914 

918 

923 

928 

933 

938 

942 

947 

952 
999 
96047 

095 
142 
190 

237 
284 
332 
379 

957 

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052 
099 

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194 

242 
289 
336 

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104 

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199 

246 
294 
341 

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06  1 
109 
156 
204 

251 
298 
346 

971 
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066 
114 
161 
209 
256 
303 
35° 
398 

976 

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671 

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1  66 
213 
261 
308 

355 

980 

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076 
123 
171 
218 
265 
313 
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080 

128 

175 
223 

270 
317 

412 

990 
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085 

133 
180 
227 

273 
322 
369 
417 

993 

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090 

'37 
183 
232 

280 

327 
374 
421 

384 

388 

393 

402 

407 

426 

473 
520 

567 
614 

66  1 
708 

755 
802 

43« 

478 

523 
572 
619 
666 

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759 
806 

435 
483 
530 

577 
624 
670 
717 
764 
811 

440 

487 
534 
58i 
628 

675 
722 

769 

816 

443 
492 

539 
586 

633 
680 

727 
774 
820 

430 
497 
544 

59i 

638 
685 

73i 

778 
825 

454 
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548 

595 
642 
689 
736 
783 
830 

459 
506 

553 
600 

647 
694 

£ 

834 

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5" 

558 
605 
652 
699 
745 
792 
839 

468 

515 
562 

609 
656 
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797 
844 

848 

453 

858 

862 

867 

872 

876 

88  1 
928 
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067 
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160 
206 
253 
299 

886 

890 

893 
942 
988 

97033 
08  1 
128 

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220 
267 

900 
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039 
086 
132 
179 

225 
271 

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137 
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192 
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285 

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104 
151 
197 

243 
290 

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109 
155 
202 
248 
294 

932 
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118 
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211 

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123 
169 
216 
262 
308 

313 
359 
405 
451 
497 

in 

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727 
772 

3£7 

322 

327 

373 
419 

463 
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557 
603 

649 

693 
740 

33i 

377 
424 
470 

516 

8 

653 
699 

743 

336 

340 

345 

330 

354 

364 
410 
456 
502 
548 
594 
640 
685 
73i 

368 
414 
460 
506 
552 
598 
644 
690 
736 

382 
428 
474 
520 
566 
612 
658 
704 
749 

387 
433 
479 
523 
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617 

663 
708 

754 

39i 

437 
483 
529 
575 
621 

667 
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759 
804 

396 
442 
488 

626 
672 
717 
763 

400 
447 
493 
539 
583 
630 
676 
722 
768 

777 

782 

786 

791 

795 

800 

809 

813 

N, 

L.  0 

I 

2 

3 

4 

5    6  |  7 

8    9 

P 

.  P. 

S.'  T.' 
9'  6.46373  373 
i°     373  373 

S."  T." 
0  15'=  900''  4-68557  558 

16  =  960      557  558 

°34'=92 

35  =93 

S."  T." 
40"  4.68  543  587 
oo      543  587 
60      543  587 
20      542  588 
80      542  588 
40      542  588 

90     368  383 
9i     368  383 
92     367  383 
94     367  383 
95     367  384 

30  =  9000      544  585 
31  =  9060      544  585 
32  =9120      543  586 
33  =  9180      543  586 
34  =  9240      543  587 

36-93 
37  =  94 
38  =  94 
39  =  95 

415 


N. 

L.  0  |   1 

234 

56789 

P 

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95O 

951 

952 

953 
954 
955 
956 
957 
958 
959 
960 
961 
962 
963 
964 
965 
966 
967 
968 
969 
970 
971 
972 
973 
974 
975 
976 

977 
978 

979 
980 

981 
982 
983 
984 
985 
986 

987 
988 
989 
990 

991 
992 
993 
994 
995 
996 

997 
998 

999 
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97772   777 

782 

786 

791 

795 

800 

804 

809 

813 

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2 

3 
4 
5 
6 

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9 

2 

3 
4 

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2-5 

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0.4 
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1.2 

1.6 

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3-6 

818 
864 
909 

955 
98000 
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091 

137 
182 

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050 
096 
141 
1  86 

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873 
918 

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146 
191 

832 
877 
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059 
105 
150 
195 

836 
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928 

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019 
064 
109 
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200 

84I 
886 
932 
978 
023 
068 
114 

159 
204 

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891 
937 
982 
028 
073 
118 
164 
209 

850 
896 
941 
987 
032 
078 
123 
1  68 
214 

855 
900 
946 
991 

037 
082 
127 
173 

218 

859 
905 
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996 
041 
087 
132 
177 
223 

227 

232 

236 

241 

245 

250 

254 

259 

263 

208 

272 
3'8 
363 
408 
453 
498 

543 
588 
632 
677 
722 
767 
811 
856 
900 
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989 
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123 
167 

211 
255 
300 

344 
388 

432 
476 
520 

277 
322 
367 
412 

457 
502 

547 
592 
637 
682 

281 
327 
372 

417 
462 

5°7 
552 
597 
641 

286 
331 
376 
421 
466 
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556 
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646 

290 
336 
381 
426 

$ 

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650 

295 
340 
385 
430 
475 
520 

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610 

655 
700 

299 
345 
39° 

435 
480 

525 
570 
614 
659 

304 
349 
394 
439 

484 
529 

574 
619 
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354 
399 

444 
489 

534 

579 
623 
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313 

358 
403 
448 
493 
538 

583 
628 
673 

686 

691 

695 

704 

709 

717 

726 

III 
860 
905 
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994 
038 
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776 
820 
865 
909 
954 
998 
043 
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735 
780 

825 
869 
914 
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047 
092 

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874 
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096 

744 
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834 
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923 
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749 
793 
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883 
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061 
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149 

753 
798 
843 
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932 
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065 
109 

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198 

242 
286 

330 
374 
419 

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802 

847 
892 
936 
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114 

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807 
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896 
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118 

127 

131 

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140 

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158 
202 

247 
291 

335 
379 
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467 
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162 

171 
216 
260 

304 
348 
392 
436 
480 
524 

176 
220 
264 
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396 
441 

484 
528 
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180 
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269 

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357 
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489 

533 

577 

185 
229 

273 
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361 
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449 
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537 

189 
233 
277 
322 
366 
410 

454 
498 
542 

193 
238 
282 
326 
370 
414 

458 
502 
546 

207 
251 
295 
339 
383 
427 

471 

515 

559 

564 
607 
651 
695 
739 
782 
826 
870 
913 
957 

568 

58i 

585 

590 

594 

599 

603 

612 
656 
699 

743 
787 
830 

874 
917 
961 

616 

660 
704 

747 
791 

835 
878 
922 
965 

621 
664 
708 

752 
795 
839 
883 
926 
97° 

625 
669 
712 

756 
800 

843 
887 
930 
974 

629 

673 
717 

760 
804 
848 
891 
935 
978 

634 
677 
721 

852 
896 
939 
983 

638 
682 
726 

3 

8S6 
900 

944 
987 

642 
686 
730 
774 
817 
861 

904 
948 
99i 

647 
691 
734 

778 
822 
865 
909 
952 
996 

oo  ooo 

004 

009 

013 

017 

022 

026 

030 

035 

039 

N. 

L.  0 

1    2    3 

4 

5 

6 

7    8  |  9 

P.  P. 

S.'   T.' 
9'  6.46  373   373 
10     373   373 

0° 

S."  T." 
15'=  900"  4-68  557  558 
[6  =  960     557  558 
[7=1020     557  558 

2°  41'=  9 

2  42  =  9 

S."  T." 
660"  4.68542  589 
720      541  590 
780      541  590 
840      541  590 
900      541  591 
960      541  591 

020        540  592 

95     367   384 
98     367   384 
99     367   385 
100     366  385 

2  4 

^3  -  9 
\4  =  9 
[5=9 
fi=  9 
17  =ic 

38  =9480      542   588 
39  =9540      542   588 
40  =9600      542   589 
41  =9660      542   589 

2  t 
2  J. 
2  i 
2  i 

416 


N. 

L.   0    1 

2     3 

4 

5 

6     7   |   8 

9 

IOOO 

IOOI 
IOO2 

1003 
1004 
1005 
1006 
1007 
1008 
1009 

IOIO 

ion 

1012 

1013 
1014 
1015 
1016 
1017 
1018 
1019 

1020 

1021 
1022 
1023 
1024 
IO25 
1026 
IO27 
1028 
IO29 

IO3O 

1031 
1032 
i°33 
1034 
1035 
1036 

1037 
1038 
1039 

1040 

1041 
1042 
1043 
1044 

1045 
1046 

1047 
1048 
1049 

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OOQ  0000    0434 

0869 

1303 

1737 

2171 

2605   3039 

3473   3907 

4341 
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oor  3009 

7337 
002  1661 
5980 
003  0295 
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9111 
3442 
7770 
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5208 
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5607 

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2882 
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7377 
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6039 
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7620 
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6472 
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9432 

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5548 
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005  1805 
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006  0380 
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007  3210 
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008  1742 

7941 
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5317 
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5201 

56l4 

N. 

L.   0 

1 

2 

3 

4 

5 

6 

789 

S."    T."                       S."    T." 
2°  46'  =  9960"    4.68541    591      2°  51'  =  10260"    4.68540    593 
2  47  =  10020        540    592      2  52  =  10320        539    594 
2  48  =  10080       540    592     2  53  =  10380       539    594 
2  49  =  10140        540    592      2  54  =  10440        539    595 
2  50  =  10200       540    593     2  55  =  10500       539    595 

41T 


N. 

L.   0 

1     2   |   3   |   4 

5     6   |   7   |   8     9 

IO5O 

1051 
1052 
i°53 
1054 
I055 
1056 

1057 
1058 
1059 
1060 
1061 
1062 
1063 
1064 
1065 
1066 
1067 
1068 
1069 

IO7O 

1071 
1072 
1073 
1074 

1075 

1076 

1077 
1078 
1079 

1080 

1081 
1082 
1083 
1084 
1085 
1086 
1087 
1088 
1089 
IO9O 
1091 
1092 
i°93 
1094 

i°95 
1096 

1097 
1098 
1099 
MOO 

02  1  1893 

2307 

2720 
6854 
0983 
5109 
9230 
3348 
7462 

1572 

5678 
9780 

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7680 
1808 
5933 
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4171 
8284 

2393 
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396i 

4374 

4787 
8919 
3046 
7170 
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5201  i  5614 

6027 
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023  2525 
6639 
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4857 
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025  3°59 

6440 
0570 
4696 
8818 
2936 
7050 
1161 
5267 
9370 

7267 
1396 

5521 
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3759 
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1982 
6088 
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8093 

2221 

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4582 
8695 
2804 
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8506 
2634 
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3214 
7319 
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9332 
3459 
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5817 
9928 
4036 
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9745 
3871 
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6228 
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4446 
8549 

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3468 

3878   4288   4697 

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5926 

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026  1245 
5333 
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027  3496 
7572 
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9824 
3904 
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6119 
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8382   8791 
2472  1  2881 
6558   6967 
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4719   5127 
8794   9201 
2865  |  3272 
6932   7339 

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9200 
3289 
7375 
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3679 

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3698 
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4297 
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4279 
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7849 

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0390173 
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040  2066 
6023 
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041  3927 

8646 
2624 
6599 
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9044 
3022 
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2858 
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1761 

5727 
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2158 
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4045 
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5009 
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295  * 
6917 

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4837 
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5407 
9379 
3348 
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5232 
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5900 

6295 

6690 

7084  |  7479 

N. 

L.   0 

1     2 

3 

4 

5 

6 

7 

8     9 

S."    T."                       S.'1    T." 
2°  55'  =  10500"   4.68539    595     3°o'=io8oo"    4.68538    597 
2  56  =  10560       539    595     31=  10860        537    598 
2  57  =  10620       538    596     32=  10920        537    598 
2  58  =  10680       538    596     33=  10980        537    599 
2  59  =10740       538    597     3  4  =11040        537    599 

K'.M'U  SLKV.  — '2~i 


418 


i 

M,  |  S'.  T', 

Sec. 

S".  T". 

0 

2 

3 
4 

1 
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9 
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12 
13 

H 
15 

16 

17 
18 

19 
20 

21 
22 
23 
24 

3 

27 
28 

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32 
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37 
38 
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4i 
42 
43 
44 

11 

47 
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49 
50 

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52 
53 
54 
55 
56 
57 
58 
59 
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1  80 
181 
182 
183 
184 
185 
186 

187 
1  88 
189 
190 
191 
192 
193 
194 

'95 
196 

199 

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353 

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412 

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68 
597 

353 
352 
352 

352 
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35  i 
35  i 
35i 
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413 
413 
414 

414 
4i5 
415 

415 
416 
416 

10860 
10920 
10980 
11040 

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11160 

1  1  220 
II280 
II340 

537 
537 
537 
537 
537 
536 
536 

53^ 
536 

^35_ 
535 
535 
535 
534 
534 
534 
534 
533 
J33. 
_533_ 
533 
532 
532 
532 
532 
53i 
53i 
53i 
53i 

598 
598 
599 
599 
599 
600 

600 
601 
601 
602 
602 
603 
603 
604 
604 
605 
605 
606 
606 

417 

II400 

350 
350 
350 
350 
349 
349 
349 
349 
348 

417 
418 
418 
419 
419 
420 
420 
421 
421 

11460 
II520 
11580 
11640 
II7OO 
11760 
II820 
II880 
II940 

200 

348 
348 
348 
347 
347 
347 
347 
346 
346 
346 

422 

12000 

607 
607 
608 
608 
609 
609 
610 
610 
611 
611 

201 
202 
203 
204 
205 
206 
207 
208 
209 

422 
423 
423 

424 
424 
425 

425 
426 
426 

427 

12060 
I2I2O 
I2I80 
12240 
I23OO 
12360 
12420 
12480 
12540 

210 

346 

12600 

530 
530 
530 
530 
529 
529 
529 

529 
528 
528 
528 

612 

211 
212 
213 
2I4 
215 

216 

217 

218 

219 

345 
345 
345 
345 
344 
344 
344 
344 
343 
343 

427 
428 
428 
429 
429 
430 
430 
43i 
43i 

12660 
12720 
12780 
12840 
I290O 
12960 
I3O20 
13080 
I3I40 

612 
613 
613 

614 
614 
615 

615 
616 
616 
~677~ 

2  2O 

432 

13200 

221 
222 
223 
224 

225 
226 
227 
228 
229 

343 
342 
342 
342 

342 
34i 
34i 
34i 
340 

432 
433 
434 
434 
435 
435 
436 
436 
437 

13260 
13320 
13380 

13440 
13500 
13560 
13620 
13680 
13740 

528 
527 
527 

527 
526 
526 
526 
526 

^i. 
525 
525 
525 
524 
524 
524 
523 
523 
523 
522 

617 
618 
618 
619 
620 
620 
621 
621 
622 

230 

340 

437 

13800 

622 

231 
232 
233 
234 
235 
236 

237 
238 

239 

340 
340 
339 
339 
339 
338 
338 
338 
338 

438 
439 
439 
440 
440 
441 
441 
442 
443 

13860 
13920 
13980' 
14040 
I4IOO 
14160 
I422O 
14280 

I434° 

623 
623 
624 
625 
625 
626 
626 
627 
628 

240 

337 

443 

14400 

522 

628 

> 

M. 

S'.  T'. 

Sec. 

S".  T".  l 

0 

i 

2 

3 
4 
5 
6 

I 

9 
IO 

ii 

12 
13 

H 
15 

16 

1? 

19 
20 

21 
22 
23 
24 

11 

29 

30 

31 
32 
33 

34 

P 

37 
38 
39 
40 

41 
42 

43 
44 
45 
46 

47 
48 

49 
5O 

51 
52 
53 
54 

| 

58 
59 
6O 

240 

6. 
337 
337 
337 
336 
336 
336 
336 
335 
335 
335 

0 

443 

14400 

4.68 
522  628 

241 

242 
243 
244 

245 
246 

247 
248 
249 

444 
444 
445 
446 
446 
447 
447 
448 
449 

14460 
14520 
14580 
14640 
14700 
14760 
14820 
14880 
14940 

522 
522 
521 
52i 
521 
520 

520 
520 

_520_ 

5!9 

629 
629 
630 

631 
631 
632 
632 
633 
634 

250 

334 

449 

15000 

634 
635 
635 
636 

637 
637 
638 
638 
639 
640 

640 

251 
252 

253 
254 
255 
256 

257 
258 
259 

334 
334 
333 
333 
333 
332 
332 
332 
332 

45o 

450 
451 
452 
452 
453 
454 
454 
455 

15060 
15120 
15180 
15240 
15300 
15360 
15420 
15480 
15540 

519 
5»9 
5i8 

5i8 
5i8 
5J7 
5i7 
5'7 

_ni 

516 

260 

33i 

456 

15600 

261 
262 
263 
264 
265 
266 
267 
268 
269 

33i 
33i 
330 
330 
330 
329 
329 
329 
328 

456 
457 
457 
458 
459 
459 
460 
461 
461 

15660 
15720 
15780 
15840 
15900 
15960 
16020 
16080 
16140 

516 
515 

5*5 

515 
5H 
5H 
5«4 

5i3 
5"3 
5U 

641 
642 
642 

643 
644 
644 

645 
646 
646 

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270 

328  |  462 

16200 

271 
272 
273 
274 
275 
276 

277 
278 

279 

328 
327 
327 
327 
326 
326 
326 
325 
325 
"325" 

463 
463 
464 

465 
465 
466 

467 
467 
468 

16260 
16320 
16380 
16440 
16500 
16560 
16620 
16680 
16740 

512 
512 

5'2 

5" 
5" 
5" 
5io 
5io 
510 
509 

648 
648 
649 
650 
650 
651 
652 
652 
653 
654 

280 

469 

16800 

281 
282 
283 
284 
285 
286 

287 
288 
289 

324 
324 
324 
323 
323 
323 
322 
322 
321 

469 
47° 
471 
472 
472 
473 
474 
474 
475 

16860 
16920 
16980 
17040 
17100 
17160 
17220 
17280 
17340 

509 
5°9 
508 
508 
508 
5°7 
5°7 
5°7 
506 

654 

^5I 
656 

656 

657 
658 

659 
659 
660 

290 

321 

476 

17400 

506 

66  1 
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662 
663 
664 
664 
665 
666 
666 
667 
668 

291 
292 
293 
294 
295 
296 

297 
298 
299 
300 

321 
320 
320 
320 
3i9 
3i9 
3J9 
3i8 
3i8 
3i7 

477 
477 
478 
479 
479 
480 

481 

482 
482 

483 

17460 
17520 
17580 
17640 
17700 
17760 
17820 
17880 
17940 
18000 

506 
5°5 
505 
505 
504 
504 

503 
5°3 
5°3 

502 

411) 


TABLE    XVI. 


THE    LOGARITHMS 


OF  THE 


TRIGONOMETRIC    FUNCTIONS 


FOR   EACH   MINUTE. 


Formulas  for  the  Use  of  the  Auxiliaries  5  and  T, 


1.    When  a  is  in  the  first  five  degrees  of  the  quadrant : 


log  sin  a  =  log  a'  +  S.1 
log  tan  a  =  log  a'  +  T.1 
log  cot  a  =  cpl  log  tan  a. 

log  sin  a  =•  log  a"  +  S." 
log  tan  a  =  log  a"  +  T.1' 
log  cot  a  =  cpl  log  tan  a. 


log  a' 
log  a" 


log  sin  a  +  cpl  S.' 
log  tan  a  +  cpl  TV 
:  cpl  log  cot  a  +  cpl  T.' 

•  log  sin  a  +  cpl  S." 
log  tan  a  +  cpl  7'." 
:  cpl  log  cot  a  +cpl  T." 


2.    When  a  is  in  the  last  five  degrees  of  the  quadrant : 


log  cos  =log(90°-o)'  +  5.' 

log  cot  =log(90°-  a)'+  T.' 

log  tan  =  cpl  log  cot  a. 

log  cos  =  log(90°  -  a)"  +  S." 

log  cot  =  log(90°  -  a)"  +  T." 

log  tan  =  cpl  log  cot  a. 

a  =  90° -(90° -a). 


log(9O°  — a)'  =  log  cos  a  +  cpl  S.' 
=  log  cot  a  +  cpl  T. ' 
=  cpl  log  tan  o  +  cpl  T.1 

log(9O°  — o)"=  log  cos  a  +  cpl  S." 
=  log  cot  a  +  cpl  T." 
—  cpl  log  tan  a  +  cpl  T." 


420 


0° 


« 

J 

L.  Sin. 

d.       Cp\.S'.     Cpl.  T'.       L.  Tan.    c.  d.     L.  Cot. 

L.  Cos. 

0 

0 

I 

2 

3 

4 

I 

s 

9 
IO 

— 

30103 

17609 

9691 
7918 
6694 
5800 

5"5 
457° 
4139 
3779 
3476 
3218 
2997 
2802 
2633 
2483 
348 
227 
119 

021 
930 
848 

773 
704 
639 
579 
524 
472 
424 

379 
336 
297 

259 
223 
190 

158 
128 

100 

072 
046 

022 

999 
976 
954 
934 
914 
896 
877 
860 

843 
827 
812 

797 
782 
769 

755 
743 
730 

!       — 

—       i|      — 

30-3 
17609 
12494 
969* 
7918 
6694 
5800 
5"5 
457° 
4J39 
3779 
3476 

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2996 

2803 

2633 

s 

2228 
2119 

2020 
I93I 
I848 

1773 
1704 
I639 

1579 
1524 

M73 
1424 

!379 
1336 
1297 

1259 
1223 
1190 

"59 
1128 

1072 
1047 

1022 
998 
976 

955 
934 
9i5 
895 
878 

860 

843 
828 
812 

797 
782 
769 

756 
742 

730 

— 

o.oo  ooo 

60 

59 
58 
57 
56 
55 
54 
53 
52 
5i 
50 

49 
48 
47 
46 
45 
44 
43 
42 
4i 
40 
39 
38 
37 
36 
35 
34 
33 
32 
3i 
30 

27 
26 
25 
24 
23 

22 
21 
20 
19 

18 
17 
16 
15 
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13 

12 
II 

IO 

§ 

7 
6 
5 
4 
3 

2 
I 

0 

6o 

240 
300 

360 

480 
540 

6-46  373 
6.76476 
6.94  085 
7.06579 
7.16  270 
7.24  188 
7.30  882 
7.36  682 
7.41  797 

3-53  627 
3-53  627 
3-53627 
3-53  627 
3-53  627 
3-53627 

3-53  627 
3-53627 
3-53  627 
3-53  627 

3-53  627 
3-53627 
3-53  627 
3-53  627 
3-53627 
3-53627 
3-53627 
3-53627 
3-53627 

6.46  373 
6.76476 
!  6.94  085 

i  7-06579 
7.16  270 
7.24  188 

;  7.30  882 
i  7-36  682 

7-4i  797 
7-46  373 
i  7-50512 
7.54291 
7-57  767 
7.60  986 
!  7.63  982 
7.66  785 
!  7.69418 
7.71  900 
i  7.74  248 

3-53627 
3-23  524 
3-05915 
2.93421 
2.83  730 
2.75812 
2.69  118 
2.63318 
2.58  203 

0.00000 
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0.00000 
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600 

746  373 

3-53  627 

2.53627 
2.49488 

2-45  7°9 
2.42  233 
2.39014 
2.36018 
2.33215 
2.30  582 
2.28  loo 

2.25  752 
2.23  524 

O.OO  OOO 

660 
720 
780 
840 
900 
960 

1020 
I080 
1140 

ii 

12 
13 
14 

\l 

17 

18 
19 

7-5°  512 
7.54291 

7-57  767 
7.60985 
7.63  982 
7.66  784 
7.69417 
7.71  900 
7.74  248 

3-53  627 
3-53627 
3-53  627 
3.53628 
3-53  628 
3-53  628 
3-53  628 
3-53628 
3-53628 

3-53627 
3-53  627 
3-53  627 
3-53627 
3-53627 
3-53627 
3-53627 
3-53627 
3-53627 

O.OO  OOO 

o.coooo 

O.OO  000 
0.00000 
O.OO  000 
0.00  OOO 

9-99  999 
9-99  999 
9-99  999 

1200 

20 

21 
22 
23 
24 

3 

27 
28 
29 

7.76475 

"7-78  594 
7.80615 
7-82  545 

7-84  393 
7.86  166 
7.87  870 
7.89  509 
7.91  088 
7.92612 

3-53  628 

3-53627 

i  7-76476 

7-78  595 
7.80615 
i  7-82  540 

7-84  394 
7.86  167 
7.87871 
i  7.89510 
7.91  089 

:  7-92613 

9.99  999 

1260 

I38o 

1440 
1500 

1560 
1620 
1680 
1740 

3-53  628 
3.53628 
3-53  628 
3.53628 
3-53  628 
3-53  628 
3-53  628 
3-53628 
3-53  628 

3-53  627 
3-53627 
3-53627 
3-53627 
3-53  627 
3-53  627 
3.53626 
3-53  626 
3-53  626 

2.21  405 
2.19385 
2.17454 
2.15  606 

2.13833 
2.12  129 

2.IO490 

2.08  9  1  1 

2.07  387 

9.99  999 
9-99  999 
9-99  999 
9-99  999 
9-99  999 
9-99  999 
9  99  999 
9-99  999 
9-99  998 

1800 

1860 
1920 
1980 
2040 

2160 

2220 
2280 
2340 

30 

3i 
32 
33 
34 

P 

s 

39 

7-94  084 

3-53  628 

3-53  626 

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8.69  708 
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254 
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251 
249 
248 
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241    239    237    236    234 

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8-73  303 

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120.5  "9-5  "8.5  118.0  117.0 

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8-73  535 

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168.7  167.3  i65-9  165.2  163.8 

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8.73  767 
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8-73  832 
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8.74  226 

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232     231     229     227     226 

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8-74454 

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1.25  252 
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8.75  130 
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208.8  207.9  206.1  204.3  2°3-4 
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29.4      29  2      29.0      28  8 

4 

8.94  603 

8-94  773 

1.05  227 

9-99  830 

56 

3 

44.1        43.8       43  5       43.2 

I 

8.94  746 
8.94  887 

141 

8.94917 
8.95  066 

143 

i  .05  083 
1.04940 

9-99  829 
9.99  828 

55 
54 

5 

6 

73  5       73  o      72  5       72  o 
88.2       87.6      87.0      864 

I 

8.95  029 
8.95  170 

142 
141 

8.95  202 

8-95  344 

142 

142 

1.04798 
1.04656 

9-99  827 
9.99  825 

53 

I 

102.9     102.2     101.5     100.8 
117.6     116.8     116.0     115.2 

9 

8.95  310 

140 

8-95  486 

142 

1.04514 

9.99  824 

51 

IO 

8-95  45° 

8-95  627 

1-04373 

9.99  823 

5O 

143        142          141        14O 

12 

8-95  589 
8-95  728 

139 

8.95  767 
8-95  908 

141 

1.04233 
1.04092 

9.99  822 
9.99821 

49 
48 

2 

3 

143       14.2       14.1       14.0 
28.6      28  4       28.2       28.0 
42.9      426      423       42.0 

13 

8.95  867 

!39 

8.96  047 

'39 

1-03953 

9.99  820 

47 

4 

57.2       56.8       56.4       560 

14 

8.96  005 
8.96  143 
8.96  280 

138 
'37 

8.96  187 
8.96  325 
8.96  464 

140 

138 

i  28 

1.03813 
1-03  673 
1.03  536 

9.99819 
9.99817 
9.99816 

46 
45 
44 

i 

I 

71.5       71.0      70.5       70.0 
85.8       852       846      84.0 
100.1       99.4      98.7      98.0 
114  4     113.6     112.8     112.  o 

17 

8.96417 

J37 

8.96  602 

1.03398 

9.99815 

43 

18 

8-96  553 

*3° 

8.96  739 

\*- 

1.03  261 

9.99814 

42 

139        138        137       136 

19 

8.96  689 

I  T.6 

8.96877 

11.6 

1.03  123 

9.99813 

41 

13.9     13.8     137     13.6 

2O 

21 

8.96  823 

135 

8.97013 

137 

1.02  987 

9.99812 

4O 
39 

3 
4 

27.8     27.6     27.4     27.2 
41.7     41.4     41.1     40.8 
55.6      55.2      54.8      54.4 

8.96  960 

8.97  130 

1.02  850 

9.99  810 

22 

8-97  °95 

J35 

8.97  285 

X35 

I.027I3 

9-99  809 

J8 

5 

69-5       690      68.5       68.0 

23 

8.97  229 

J34 

8.97421 

X36 

1.02579 

9.99  808 

37 

24 

8-97  363 

*34 

8-97  556 

J35 

1.02444 

9-99  807 

J6 

8 

in.  2     110.4     109.6     108.8 

25 

8.97  496 

Jjj 

8.97  691 

'35 

1.02309 

9.99  806 

35 

9 

125.1     124.2     123.3     122.4 

26 

8.97  629 

M3 

8.97  823 

X34 

1.02  175 

9.99  804 

34 

27 

8.97  762 

J33 

8-97  959 

J34 

I.0204I 

9-99  803 

33 

28 

8.97  894 

132 

8.98092 

J33 

i.oi  908 

9.99  802 

3 

27.0      26.8      26.6      26.4 

29 

8.98  026 

132 

8.98  225 

J33 

i.oi  773 

9.99  80  1 

31 

3 

40.5      40.2      39.9      39.6 

30 

8.98  157 

8.98  358 

1.01642 

9.99  800 

30 

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67.5      67.0      665      66.0 

31 

8.98  288 

8.98  490 

i.oi  510 

9-99  798 

2Q 

6 

81.0      80.4      798      79.2 

32 

33 

8.98419 
8.98  549 

130 

8.98622 
8.98  753 

132 
13* 

i.oi  378 
i.oi  247 

9-99  797 
9.99  796 

28 
27 

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94-5       93-8       93-i       924 
108.0     107.2     106.4     105.6 
121.5     120.6     119.7     u8.8 

34 

8.98  679 

130 

8.98  884 

I3I 

i.oi  116 

9-99  795 

2(> 

35 

8.98  808 

129 

8.99015 

I3I 

1.00985 

9-99  793 

2^ 

131         130        129        128 

36 

8.98  937 

129 

8-99  J45 

130 

1.00853 

9-99  792 

24 

1 

13.1       13.0       12.9       12.8 

37 

8.99066 

129 

8.99  275 

130 

i.oo  723 

9  99  791 

23 

3 

39-3       39-o      38-7       38-4 

38 
39 

8-99  194 
8.99  322 

128 
128 

8.99  403 
8-99  534 

130 
129 
128 

1.00595 

1  .00  466 

9-99  790 
9-99  788 

22 
21 

4 
5 

6 

52.4       52.0      51.6       51.2 
65.5       65.0      64.5      64.0 
78.6       78.0      77.4       76.8 

4O 

8.99  430 

8.99  662 

i.oo  338 

9.99  787 

20 

I 

91.7      91.0      90.3       89.6 

41 

42 

8-99  577 
8.99  704 

127 

8.99  791 
8.99919 

128 

1  .00  209 
1.00081 

9-99  786 
9-99  785 

19 
18 

9 

117.9     "7-°     "o-1     JI5-2 

43 

8.99  830 
8.99  956 

126 

9.00  046 
9.00  174 

127 

128 

0.99  954 
099  826 

9-99  783 
9.99  782 

17 
1  6 

127         126        123        124 

45 

9.00082 

126 

9.00301 

127 

0.99  699 

9-99  78i 

15 

25.4      25.2       25.0      24.8 

46 

9.00  207 

125 

9.00427 

0-99  573 

9  99  780 

14 

4 

50.8       50.4       500      496 

47 

9.00  332 

125 

9.00  553 

126 

0.99  447 

9-99  778 

i.S 

63  5       63  o      62  5      62  o 

48 

9.00456 

124 

9.00  679 

0.99  321 

9  99  777 

12 

7 

88.9      88.2      87.5      868 

49 

9.00  581 

125 

9.00  803 

0.99  19  s 

9-99  776 

II 

S 

101.6     100.8     loo  o      99.2 

50 

9.00  704 

9.00  930 

I25 

0.99  070 

9-99  775 

10 

9 

114.3     "3-4     1125     "1-6 

51 

9.00  828 

9.01  055 

I25 

0.<>S  945 

9-99  773 

9 

123         122         121         120 

52 

9.00951 

9.01  179 

124 

0.98821 

9-99  772 

8 

12  3          12.2          12.  1          12  0 

53 

9.01  074 

'-3 

9.01  303 

0.98  697 

9.99771 

7 

j 

24.6         24.4         242         24.0 

54 
55 

9.01  196 
9.01  318 

122 

122 

9.01  427 
9.01  550 

124 
123 

0.98  573 
0.98  430 

9-99  769 
9-99  768 

5 

3 
4 
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36.9          36.6         36.3          36.0 
49-2         48.8         484         48.0 

61.5      61.0      60.5      60.0 

56 

57 

9.01  440 
9.01  561 

121 

9.01  673 
9.01  796 

123 

0.98  327 
0.98  204 

9-99  767 
9-99  765 

4 

0 

I 

73-8       73-2       72-6       72-° 
86.1       85-4       847       84.0 
98.4      97-6      96.8      96.0 

58 

9.01  682 

9.01  918 

0.98  082 

9-99  764 

2 

9 

1,0.7     109-8     1089     108.0 

59 

9.01  803 

9.02  040 

0.97  960 

I 

6O 

9.01  923  j 

9.O2  102 

0.97  838 

9.99761 

0 

L.  Cos.  |   d. 

L.  Cot.  |c.  d.    L.  Tan. 

L.  Sin. 

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P.  P. 

84° 


426 


i  / 

L.  Sin. 

d. 

L.  Tan. 

c.d. 

L.  Cot. 

L.  Cos. 

P.  P. 

o 

9.01  923 

9.02  162 

0.97  838 

9.99  761 

60 

I 

2 

3 

4 

I 

9 

9.02  043 
9.02  163 
9.02  283 
9.02  402 
9.02  520 
9.02  639 
9.02  757 
9.02  874 
9.02  992 

120 
120 
119 

118 

119 

118 
117 
118 

9.02  283 
9.02  404 
9-02  525 

9.02  645 

9.02  766 
9.02  885 
9.03  005 
9.03  124 
9.03  242 

121 
121 
1  2O 
121 
119 
120 
119 

118 

0.97717 
0.97  596 
0-97  475 
0-97  355 
0-97  234 
0.97115 

0.96  995 
0.96  876 
0.96  758 

9.99  760 
9-99  759 
9-99  757 
9-99  756 
9-99  755 
9-99  753 
9-99  752 
9-99  75  » 
9-99  749 

59 
5* 
57 
56 
55 
54 
53 
52 
51 

I 
a 
3 
4 

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121    12O    119    118 
12.  1   12.0   11.9   ii.  8 
24.2   24.0   23.8   23.6 
36.3   36.0   35.7   35.4 
48.4   48.0   47.6   47.2 
60.5   60.0   59.5   59.0 
72.6   72.0   71.4   70.8 

IO 

9.03  109 

9.03361 

1  18 

0.96  639 

9.99  748 

50 

I 

ii 

12 
13 

14 

9.03  226 
9-03  342 
9.03  458 

9-03  574 
9-03  690 
9.03  803 
9.03  920 

117 
116 
116 
116 
116 
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9-03  479 
9-03  597 
9.03  714 

9.03  832 
9.03  948 
9.04  065 
9.04  181 

118 
117 
118 
116 
117 
116 

0.96  521 
0.96  403 
0.96  286 
0.96  1  68 
0.96052 
0-95  935 
o  95  819 

9-99  747 
9-99  745 
9-99  744 
9.99  742 
9-99  74i 
9.99  740 

9  99  7  "?8 

49 
48 

47 
46 
45 
44 

9 

108.9  108.0  107.1  106.2 

117    116    115    114 

11.7   ii.  6   11.5   11.4 

19 

9.04  034 
9.04  149 

114 
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9.04  297 
9.04413 

116 
116 

0.95  703 
0-95  587 

9-99  737 
9-99  736 

42 

3 
4 
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35.1   34.8   34.5   34.2 
46.8   46.4   46.0   45.6 
58.5   58.0   57.5   57.0 

20 

9.04  262 

9.04  528 

0.95  472 

9-99  734 

40 

6 

70.2   69.6   69.0   68.4 

21 
22 
23 

24 

3 

27 
28 
29 

9.04  376 
9.04  490 
9.04  603 

9-°4  71  5 
9.04  828 
9.04  940 
9.05052 
9.05  164 
9-05  273 

114 
113 

112 

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112 
112 
112 
III 

9.04  643 
9.04  758 
9.04  873 
9.04  987 
9.05  101 
9.05  214 
9.05  328 
9.05  441 
9-05  553 

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114 
114 
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114 

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112 

0-95  357 
0-95  242 
0.95  127 

0.95013 
0.94  899 
0.94  786 
0.94  672 
0-94  559 
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9-99  733 
9-9973' 
9-99  730 
9-99  728 
9.99  727 
9-99  726 
9.99  724 
9-99  723 
9.99721 

39 
38 

37 
36 
35 
34 
33 
32 

8 
9 

3 

93.6   92.8   92.0   91.2 
105.3  104.4  103.5  102.6 

113    112    111    110 

II.  3     II.  2     II.  I     II.  0 
22.6    22-4    22.2    22.0 

33-9   33-6   33-3   33-O 

30 

9.05  386 

9.05  666 

o-94  334 

9.99  720 

3O 

5 

56.5   56.0   55.5   55.0 

32 
33 
34 

I 

39 

9-05  497 
9.05  607 
9.05  717 
9.05  827 
9-05  937 
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9-o6  155 
9.06  264 
9.06  372 

IIO 
IIO 
IIO 
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109 
109 
1  08 

9.05  778 
9.05  890 
9.06  002 
9.06  113 
9.06  224 
9-o6  333 
9.06  445 
9.06  556 
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112 
112 
III 
III 
III 
IIO 
III 
IIO 

0.94  222 

0.94  110 

0.93  998 

o-93  887 
0.93  776 
0.93  665 

0-93  555 
0.93  444 

0-93  334 

9-99  7J8 
9.99717 

9-99  7l6 
9-99  7H 
9-99  7'3 
9.99711 

9.99  710 
9-99  7°8 
9-99  7°7 

29 

28 

27 
26 
25 
24 
23 

22 
21 

6 

I 
9 

2 

67.8   67.2   66.6   66.0 
79.1   78.4   77.7   77.0 
90.4   89.6   88.8   88.0 
101.7  100.8   99.9   99.0 

109   108    107   106 
10.9   10.8   10.    10.6 

21.8    21.6    21.      21.2 

40 

9.06  48  1 

1  08 

9.06  775 

0.93  225 

9.99  705 

2O 

3 

32.7  32-4  32-   31-8 

41 
42 
43 
44 
45 
46 

47 
48 

49 

9.06  589 
9.06  696 
9.06  804 
9.06911 
9.07018 
9.07  124 
9.07  231 
9-07  337 
9.07  442 

107 
1  08 
107 
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106 
107 
1  06 

9.06  8&5 
9.06  994 
9.07  103 

9.07  211 
9.07  320 
9.07  428 
9.07  536 
9.07  643 
9.07751 

IIO 

109 
109 

108 
109 
108 
108 
107 
108 

o.93"5 
0.93  006 
0.92  897 
0.92  789 
0.92  680 
0.92  572 
0.92  464 

0.92357 
0.92  249 

9-99  7°4 
9-99  702 
9.99  701 

9-99  699 
9.99  698 
9-99  696 
9.99  695 

9-99  693 

9.99  092 

19 

18 
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16 
15 

13 
ii 

1 

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9 

54-5   54-o   53-    53-O 
65.4   64  8   64.    63.6 
76.3   75.6   74.9   74  2 
87.2   86  4   85.6   84.8 
98.1   97.2   96.3   95.4 

105    104    103 

5O 

9.07  54S 

9.07  858 

107 
1  06 

0.92  142 

9.99  690 

IO 

2   21        208      20.6 

51 
52 
53 
54 

P 

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59 

9-0?  653 
9.07  758 
9.07  863 
9.07  968 
9.08  072 
9.08  176 
9.08  280 
9.08  383 
9.08  486 

I05 
105 
I05 

104 
104 
104 
103 
103 

9.07  964 
9.08071 
9.08177 
9.08  283 
9.08  389 

9.08  493 
9.08  600 
9.08  705 
9.08810 

107 
106 
106 
106 
106 

I05 
105 

0.92  036 
0.91  929 
0.91  823 
0.91  717 
0.91  611 
0.91  505 
0.91  400 
0.91  293 
0.91  190 

9.99  689 
9.99  687 
9.99  686 
9-99  684 
9-99  683 
9.99  68  1 
9.99  680 
9.99  678 
9-99  677 

9 
8 

7 
6 
5 
4 
3 

2 
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3  1  31     31-2    3°-9 
4  42     41.6    41.2 

6  63     62.4   61.8 
7  73-5    72.8   72.1 
8  84.0   83.2   82.4 
9  94-5    93-6    927 

6O 

9.08  589 

9.08914 

0.91  086 

9-99  675 

O 

L.  Cos. 

d. 

L.  Cot. 

c.d. 

L.  Tan. 

L.  Sin. 

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P.  P. 

83s 


L.  Sin. 

d. 

L.  Tan.  c.d.  L.  Cot. 

L.  Cos. 

p.  p. 

0 

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2 

3 

4 

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7 
8 
9 
IO 
ii 

12 
13 
14 

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17 

18 
19 
20 

21 
22 
23 
24 
25 
26 

27 
28 
29 

30 

3i 
32 
33 
34 

3 

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39 
40 

4i 

42 
43 
44 

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47 
48 
49 
50 

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52 
53 
54 
55 
56 

H 

59 
60 

9.08  589 
9.08  692 
9.08  795 
9.08  897 
9.08  999 
9.09  101 

9.09  202 
9.09  304 
9.09  405 
9.09  506 

103 
103 
1  02 

102 
102 
101 
IO2 
101 
101 
100 
IOI 
100 
100 

99 

100 

99 
99 

98 

98 
98 
98 
97 
97 
97 
97 
96 

97 
96 

96 

95 
96 

95 
95 
95 
94 
95 
94 
94 
93 
94 
93 
93 
93 
93 
93 
92 
92 
92 
92 

9i 
92 

9i 
9i 
90 

9i 
90 

91 
90 

9.08914 

I05 
104 
104 
103 
104 
103 
103 

102 
103 
IO2 
IO2 
IOI 
IO2 
IOI 
IOI 
JOI 
IOI 
IOO 
IOO 
IOO 
IOO 

99 
99 
99 
99 
99 
98 
98 
98 
98 

97 
98 
97 
97 
96 
97 
96 
96 
96 
96 

95 
95 
95 
95 
95 
94 
95 
94 
94 
93 
94 
93 
93 
93 
93 
92 
92 
93 
9i 

92 

0.91  086 

9-99  675 

60 

59 
58 
57 
56 
55 
54 
53 
52 
5' 
50 
4<> 
48 
47 
4" 
45 
44 
43 

4-2 

4i 
40 
39 
38 
37 
36 
35 
34 

33 

32 
31 
30 

29 
28 
-1 
26 
25 
24 
23 

22 
21 

20 

19 

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13 

12 

n 
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9 
8 

7 
6 
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4 
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2 
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7 

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9 

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2 

3 

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7 
8 
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2 

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105   104   103 
10.5   10.4   10.3 

21.0    20.  8    2O.6 

31.5  31.2  30.9 

42.0   41.6   41.2 

52.5  52.0  51.5 
63.0  62.4  61.8 
73.5  72.8  72.1 
84.0  83.2  82.4 

94-5   93-6   92-7 
102   101    99 

10.2    IO.I     9.9 
20.4    20.2    19.8 
30.6    30.3    29.7 
40.8    40.4    39.6 
51.0    50.5    49.5 
6l.2    60.6    59.4 
71.4    70.7    69.3 
8l.6    80.8    79.2 
91.8    90.9    89.1 

98    97    96 

9.8    9.7    9.6 
19.6   19.4   19.2 
29.4   29.1   28.8 
39.2   38.8   38.4 
49.0   48.5   48.0 
58.8   58.2   57.6 
68.6   67.9   67.2 
78.4   77.6   76.8 
88.2   87.3   86.4 

95    94    93 

9-5    9-4    9-3 
19.0   18.8   18.6 
28.5   28.2   27.9 
38.0   37.6   37.2 
47.5   47.0   46.5 
57-o   56-4   55-8 
66.5   65.8   65.1 
76.0   75.2   74.4 
85.5   84.6   83.7 

92    91    90 

9.2    9.1    9.0 
18.4   18.2   18.0 
27.6   27.3   27.0 
36.8   36.4   36.0 
46.0   45.5   45.0 
55.2   54.6   54.0 
64.4   63.7   63.0 
73.6   72.8   72.0 
82.8   81.9   81.0 

9.09019 
9.09  123 
9.09  227 
9.09  330 
9-09  434 
9-09  537 
9.09  640 
9.09  742 
9.09  845 

0.90981 
0.90  877 
0.90  773 
0.90  670 
0.90  566 
0.90  463 
0.90  360 
0.90  258 
0.90  155 

9-99  (J74 
9-99  672 
9-99  670 
9-99  669 
9-99  667 
9.99  666 

9-99  664 
9-99  663 
9.99661 

9.09  6oO 

9.09  947 

0.90053 

9-99  659 

9.09  707 
9.09  807 
9.09  907 
9.10006 
9.10  106 
9.10205 
9.10304 
9.10402 
9.10501 
9-10599 
9.10697 
9.10795 
9.10893 
9.10990 
9.11  087 
9.11  184 
9.11  281 

9-"  377 

9.11474 

9.10049 
9.10  150 
9.10252 

9-10353 
9.10454 
9-10555 
9.10656 
9.10756 
9.10856 

0.89951 
0.89  850 
0.89  748 
0.89  647 
0.89  546 
0.89  445 
0.89  344 
0.89  244 
0.89  144 

9-99  658 
9-99  656 
9-99  655 

9-99  653 
9.99651 
9-99  650 
9-99  648 
9-99  647 
9-99  645 

9.10956 
9.11  056 

9-"  155 
9.11354 

9-"  353 
9-II452 
9-n  551 
9.11649 
9.  ii  747 
9.11845 

0.89  044 

9-99  643 

0.88  944 
0.88  845 
0.88  740 
0.88  647 
0.88  548 
0.88  449 
0.88351 
0.88  253 
0.88  155 

9-99  642 
9.99  640 
9-99  638 
9-99  637 
9-99  635 
9-99  633 
9-99  632 
9-99  630 
9-99  629 

9.11  570 
9.11  666 
9.11  761 
9.11857 
9.11952 
9.12047 
9.12  142 
9.12  236 
9-12331 
9.12425 

9-  "943 

9.  1  2  O4O 
9.I2I38 
9.12235 
9.12332 
9.12428 
9.12525 
9-12  621 
9.12717 
9-12  813 
9-I2909 

0.88057 

9-99  627 

0.87  960 
0.87  862 
0.87  765 
0.87  668 
0.87572 
0-87  475 
0.87  379 
0.87  283 
0.87  187 

9-99  625 
9.99  624 
9.99  622 
9.99  620 
9.99  618 
9.99617 
9.99615 
9.99613 
9.99612 

9.12519 
9.12  612 
9.12  706 
9.12799 
9.12892 
9.12985 
9.13078 

9-I3I71 
9.13  263 

9-13355 

0.87091 

9.1)9  bio 

9.13004 
9.13099 
9-13  194 
9.13  289 
9-I3384 
9-I3478 

9-13573 
9.13667 
9.13761 

0.86  996 
0.86  901 
0.86  806 
0.86711 
0.86616 
0.86  522 
0.86427 
0.86333 
Q-86  239 
p.86  146 

9.99  608 
9-99  607 
9-99  605 

9-99  603 
9.99  60  1 
9.99600 

9-99  598 
9-99  596 
9-99  59? 

9-'3447 

9-I3854 

9-99  5'>3 

9-13539 
9.13630 
9.13722 
9-13813 
9.13904 
9-13994 
9.14085 

9-HI75 
9.14266 

9.13948 
9.I404I 
9.14  134 
9.14227 
9.14320 
9.I44I2 
9.14  504 

9-H597 
9.14688 

0.86052 
0-85  959 
0.85  866 

0-85  773 
0.85  680 
0.85  588 
0.85  496 
0.85  403 
0.85  312 

0.85  220 

9-99  59' 
9-99  589 
9-99  588 
9-99  586 
9-99  584 
9-99  582 
9-99  581 
9-99579 
9^9577 
9-99  575 

9-H356 

u.I.l  -S<> 

L.  Cos. 

d. 

L.  Cot.  c.d.  L.  Tan. 

L.  Sin. 

' 

P.  P. 

82' 


428 


8° 


1 

L.  Sin. 

d. 

L.  Tan. 

c.d. 

L.  Cot. 

L.  Cos. 

p.  p. 

0 

9-I4356 

80 

9.14780 

0.85  220 

9-99  575 

60 

I 

2 

3 

4 

I 

9 

9.14445 

9-14535 
9.14624 

9.14714 
9.14803 
9.14891 
9.14980 
9.15069 
9-15  157 

90 
89 
90 

89 
88 
89 

88 

9.14872 
9.14963 
9-15  °54 

9-I5  145 

9.15236 

9.15327 

9-15417 
9.15  508 

90 
90 

0.85  128 
0.85  037 

0.84  946 

0.84  855 
0.84  764 
0.84673 
0.84  583 

0.84  492 
0.84  402 

9-99  574 
9-99  572 
9-99  570 
9-99  568 
9-99  566 
9-99  565 
9-99  563 
9-99  561 
9-99559 

59 

58 
57 
56 
55 
54 
53 
52 
51 

92    91    90 

i   9.2    9.1    9.0 
2  18.4   18.2   18.0 
3  27.6   27.3   27.0 
4  36.8   36.4   36.0 
5  46.0   45.5   45.0 
6  55.2   54.6   54.0 
7  64.4   63.7   63.0 

IO 

9-15245 

88 

9.15688 

0.84312 

9-99  557 

50 

ii 

12 
13 

15 

16 
19 

9-15  333 
9.15421 
9.15  508 

9-15  596 
9.15  683 
9.15  770 

9-I5857 
9.I5  944 
9.16030 

88 
87 
88 

87 
87 
87 
87 
86 
86 

9-15777 
9.15867 
9.I5956 
9.16046 
9-16133 
9.16224 
9.16  312 
9.16401 
9.16489 

*N  O  O\  O  O\  ON  00  ONOO  0< 
3  ONOO  O  00  OO  00  00  00  0( 

0.84  223 

0.84  133 

0.84  044 
0-83  954 

0.83  865 
0.83  776 

0.83  688 
0.83  599 
0.83511 

9-99  SS6 
9-99  554 
9-99552 
9-99  55° 
9-99  548 
9.99  546 

9-99  543 
9-99  543 
9-99  54i 

49 
48 

47 
46 
45 
44 
43 
42 

89     88 

i   8.9    8.8 
2  17.8    17.6 
3  ;  26.7    26.4 
4  35-6    35-2 
5  i  44.5   44.0 

2O 

9.16  116 

87 

9.16577 

88 

0-83  423 

9-99  539 

40 

6  !  53-4    52-8 

21 
22 
23 
24 
25 
26 
27 
28 
29 

9.16  203 
9.16  289 
9-16374 
9.16460 

9-16545 
9.16631 

9.16  716 
9.16801 
9.16886 

86 

85 
86 

85 
86 

85 
85 

9.16665 
9-16753 
9.16  841 
9.16928 
9.17016 
9.17  103 
9.17  190 
9.17277 
9-I7363 

88 
88 

87 
88 
87 
87 
87 
86 

0-83  335 
0.83  247 
0-83  159 
0.83  072 
0.82  984 
0.82  897 
0.82810 
0.82  723 
0.82  637 

9-99  537 
9-99  535 
9-99  533 
9-99  532 
9-99  53° 
9-99  528 
9-99  526 
9-99  524 
9.99  522 

39 
38 

37 
36 
35 
34 
33 

3i 

8  71.2    70.4 
9  80.  i    79.2 

87    86    85 

i   8.7    8.6    8.5 
2  17.4   17.2   17.0 
3  26.1   25.8   25.5 

30 

9.16970 

Or 

9.17430 

S7 
86 

0.82  550 

9-99  520 

30 

4  34-8   34-4   34-° 

32 
33 
34 
35 
36 

38 
39 

9-I7055 
9-17  J39 
9.17223 

9.17307 
9-I739I 
9-17474 

9-I7558 
9.17641 
9.17724 

°5 
84 
84 
84 
84 
83 
84 
83 
^ 

9-I7536 
9.17  622 
9.17708 
9.17  794 
9.17880 
9.17965 
9.18051 
9.18  136 

9.18  221 

86 
86 
86 
86 
85 
86 

11 

0.82  464 
0.82  378 
0.82  292 
0.82  206 
0.82  1  20 
0.82035 
0.81  949 
0.8  1  864 
0.81  779 

9.99518 
9-99  51  7 
9.995*5 
9-99  5  J3 
9-99  511 
9-99  5°9 
9-99  5°7 
9-99  5°5 
9-99  5°3 

29 
28 
27 
26 
25 
24 
23 

22 
21 

6  52.2   51.6   51.0 
7  60.9   60.2   59.5 
8  69.6   68.8   68.0 
9  78-3   774   76-5 

84     83 

i   8.4    8.3 

40 

9.17807 

83 

Si 

9.18  306 

0, 

0.81  694 

9-99  501 

2O 

42 
43 
44 

11 

47 
48 
49 

9.17890 

9-17973 
9.18055 

9-18137 
9.18220 
9.18  302 

9-18383 
9.18465 
9-18547 

3  OOOOOOOOOOOOOOOOCX 

9.18391 
9-I8475 
9.18560 
9.18644 
9.18728 
9.I88I2 
9.18896 
9.18979 
9.19063 

84 
85 
84 
84 
84 
84 
83 
84 

Ci 

0.81  609 
0.8  1  525 
0.81  440 
0.81  356 
0.81  272 
0.8  1  1  88 
0.81  104 
0.81  021 
0.80  937 

9-99  499 
9-99  497 
9-99  495 
9-99  494 
9-99  492 
9.99  490 

9-99  488 
9.99  486 
9-99  484 

19 

18 

17 

1  6 
15 

13 

12 
II 

4  33-6    33-2 
5  42.0   41.5 
6  |  50.4   49.8 
7  58.8   58.1 
8  67.2   66.4 
9  75-6   74-7 

82    81    80 

50 

9.18628 

81 

9.19  146 

Si 

0.80  854 

9-99  482 

10 

i   82    8  i    80 

5' 

52 
53 
54 
55 
S6 

58 

59 

9.18709 
9.18790 
9.18871 
9.18952 

9-19033 
9.19113 

9.19193 
9.19273 
9-19353 

81 
81 
81 
81 
80 
80 
80 
80 
80 

9.19229 
9.I93I2 
9-19395 
9.19478 
9.19561 
9.19  643 
9.19  725 
9.19807 
9.19889 

3  OOOOOOOOOOOOOOOOOI 

0.80  771 
0.80  688 
0.80  605 
0.80  522 
0.80  439 
0.80  357 
0.80  275 
0.80  193 
0.80  1  1  1 

9-99  480 
9-99  478 
9.99476 

9-99  474 
9.99472 
9-99  47° 
9.99  468 
9.99  466 
9-99  464 

9 

8 
7 
6 
5 
4 
3 

2 

2  16.4   1  6.2   1  6.0 
3  24.6   24.3   24.0 
4  32.8   32.4   32.0 
5  41.0   40.5   40.0 
6  49.2   48.6   48.0 
7  57-4   56-7   56-0 
8  65.6   64.8   64.0 
9  ;  73.8   72.9   72.0 

6O 

9-19433 

9.19971 

0.80  029 

9-99  462 

O 

' 

L.  Cos. 

d. 

L.  Cot. 

c.d. 

L.  Tan. 

L.  Sin. 

' 

P.  P.        j 

81C 


429 


L.  Sin. 

d. 

L.  Tan. 

c.d. 

L.  Cot. 

L.  Cos. 

P.  P. 

0 

9-19433 

9.19971 

0.80  029 

9.99  462 

60 

I 

2 

3 

4 

I 

7 
8 
9 

9-i95'3 
9.19592 
9.19672 
9.19751 
9.19830 
9.19909 
9.19988 
9.20067 
9.20  145 

79 
80 

79 
79 
79 
79 
79 
78 

9.20053 
9.20  134 
9.20  216 
9.20  297 
9.20  378 
9.20459 
9-20  540 
9.20621 
9.20  701 

81 
82 
81 
81 
81 
81 
81 
80 

XT 

0.79  947 
0.79  866 
0.79  784 
0.79  703 
0.79  622 
0.79  541 
0.79  460 
0-79  379 
0.79  299 

9.99  460 
9-99  458 
9-99  456 
9-99  454 
9-99452 
9-99  450 
9.99  448 
9-99  446 
9.99  444 

!I 

57 
56 

55 
54 
53 
52 
Si 

2 

7 

82          81          80 

8.2        8.1         8.0 
16.4       16.2       16.0 
24.6       24.3       24.0 
32.8       32.4      32.0 
41.0      40.5       40.0 
49.2      48.6      48.0 
57-4      56-7      56-0 

IO 

9.20  223 

78 

9.20  782 

Sr> 

0.79  218 

9-99  442 

50 

8 

65.6      64.8      64.0 

ii 

12 
13 

H 

!S 

17 
18 

19 

9.20  302 
9.20  380 
9.20458 

9-20  535 
9.20613 
9.20691 
9.20  768 
9.20  845 
9.20922 

79 
78 
78 
77 

It 

77 

77 
77 

9.20  802 

9.20  942 

9.21  022 
9.21   IO2 
9-21   l82 
9-21  26l 
9.21  341 
9.2I  420 

9.21  499 

so 
so 

So 

80 
79 
80 
79 
79 

0.79  138 
0.79  058 
0.78978 
0.78  898 
0.78818 
0.78  739 
0.78  659 
0.78  580 
0.78  501 

9-99  44° 
9-99  438 
9-99  436 
9-99  434 
9-99  432 
9.99  429 
9.99427 
9.99425 
9-99  423 

49 
48 

47 
46 
45 
44 
43 
4- 
41 

9 

I 

2 

3 

4 

I 

73.8       72.9      72.0 

79          78         77 

7-9         7-8         7-7 
15.8       15.6       15.4 
23.7       23.4       23.1 
31.6       31.2       30.8 
39-5       39-o      38-5 
47.4      46.8      46.2 

20 

9.20999 

77 

9-21  578 

0.78422 

9.99421 

4O 

I 

55-3      54-6      53-9 

21 
22 
23 
24 

25 
26 

27 
28 
29 

9.21  076 
9.21  153 
9.21  229 
9.21  306 
9.21  382 
9.21  458 

9-2i  534 
9.21  610 
9.21  685 

77 

M 

77 
76 
76 
76 
76 
75 

9.21  657 
9-21  736 

9.21  814 
9.21  893 
9.21971 
9.22  049 
9.22  127 

9.22  205 
9.22  283 

79 
78 
79 
78 
78 
78 
78 

7* 

0.78  343 
0.78  264 
0.78  1  86 
0.78  107 
0.78029 
0.77951 
0.77  873 
0.77  795 
0.77717 

9.99419 
9.99417 
9-994I5 

9-994I3 
9.99411 
9.99  409 

9-99  4°7 
9.99  404 
9.99  402 

39 
38 

37 
36 
35 
34 
33 
32 
3i 

9 

i 

2 

3 
4 
I 

71.1       70.2      69.3 

76         75          74 

7-6        7-5        7-4 
15.2       15.0       14.8 

22.8          22.5          22.2 
30.4         30.0          29.6 

38-0      37-5       37-o 

SO 

9.21  761 

/° 

9.22  361 

7° 

0.77  639 

9.99  400 

30 

45.6      45.0      44.4 

31 
32 

33 
34 

P 

37 
38 
39 

9.21  836 
9.21  912 
9.21  987 
9.22062 

9-22  137 
9-22  211 
9-22  286 
9.22361 
9-22435 

75 
76 
75 
75 
75 
74 
75 
75 
74 

9.22  438 

9.22  516 
9-22  593 
9.22  670 
9-22  747 
9.22  824 
9.22  901 
9.22977 
9.23  054 

77 
78 
77 
77 
77 
77 
77 
76 

11 

0.77  562 
0.77  484 
0.77  407 

o-77  33° 
o-77  253 
0.77  176 

0.77099 
0.77023 
0.76  946 

9-99  398 
9-99  396 
9-99  394 
9-99  392 
9-99  390 
9-99  388 
9-99  385 
9-99  383 
9-99  38i 

29 
28 
27 
26 
25 
24 
23 

22 
21 

i 

9 
i 

2 

3 

4 

S3-2      52-5      5J-8 
60.8      60.0      59.2 
68.4      67.5      66.6 

73          72         71 

7-3        7-2        7-i 
14.6       14.4       14.2 
21.9       21.6       21.3 
29.2     28.8     28.4 

40 

9-22  509 

74 

9.23  130 

76 

0.76870 

9-99  379 

20 

f 

36.5     36.0     35.5 

4i 
42 
43 
44 

9.22583 
9.22657 
9.22731 
9-22  805 
9  22  878 

74 
74 
74 
74 

73 

9.23  206 
9.23  283 
9-23  359 
9-23435 

Q  2"?  SIO 

76 

H 

76 

75 

0.76  794 
0.76717 
0.76  641 
0.76  565 

9-99  377 
9-99  375 
9-99  372 
9-99  370 
9  99  3^8 

19 
18 
17 
16 

9 

51.1     50.4     49.7 

58.4      57-6      56-8 
65.7      64.8      63.9 

£ 

47 
48 
49 

9.22952 
9.23025 
9.23  098 
9.23  171 

74 
73 
73 
73 

9.23  586 

9.23  661 

9-23  737 
9.23812 

76 

75 
76 
75 

0.76414 

0-76  339 
0.76  263 
0.76  188 

9-99  366 
9-99  364 
9-99  362 
9-99  359 

I4 

13 

12 
II 

o 

333 

79         78         77 

13  2         13.0         12.8 

SO 

9.23  244 

73 

9-23  887 

75 

0.76  113 

9-99  357 

10 

i 

39-5      39-°      38-5 

5i 

52 
53 

54 

!i 
12 

59 

9.23317 
9.23  390 
9.23  462 

9-23  535 
9.23  607 
9.23  679 

9-23752 
9.23823 
9-23  895 

73 

73 
72 

73 
72 
72 
73 
7i 
72 

9.23  962 
9.24037 
9.24112 
9.24  1  86 
9.24  261 
9-24  335 
9.24410 
9.24484 
9-24558 

75 
75 
75 
74 
75 
74 
75 
74 
74 

0.76038 

o-75  963 
0.75  888 

0.75  814 

o-75  739 
0.75  665 

0.75  590 

o-75  5'6 
0.75  442 

9-99  355 
9-99  353 
9-99  35  * 
9-99  348 
9-99  346 
9-99  344 
9-99  342 
9-99  340 
9-99  337 

7 
6 
5 
4 

3 

2 

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3 

o 
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2 

3 

65.8      65.0      64.2 

333 

76         75          74 
12.7       12.5       12.3 
38-0      37-5       37-o 
63.3       62.5       61.7 

60 

9.23967 

9.24632 

0.75  368 

9-99  335 

0 

L.  Cos. 

d. 

L.  Cot. 

c.d. 

L.  Tan. 

L.  Sin. 

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P.  P. 

80° 


430 


10° 


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L.  Sin. 

d. 

L.  Tan.  |c.  d.     L.  Cot. 

L.  Cos. 

d. 

p.  p.          1 

0 

I 

2 

3 

4 

9 
IO 

ii 

12 
13 
H 

16 

19 
20 

21 
22 
23 
24 

3 

2 

29 
30 

3i 
32 
33 
34 

P 

11 

39 
4O 

4i 
42 

43 
44 

8 

3 

49 
50 

5i 
52 
53 
54 

P 
P 

59 
6O 

9.23  967 

9-24  039 
9.24  no 
9.24  181 
9.24  253 
9.24  324 
9-24  395 
9.24  466 
9-24  536 
9.24607 

72 
7i 
7i 

72 

7i 
7i 
7i 
70 

7i 

70 

7i 

70 
70 
70 
70 
70 
70 
69 
70 
69 
69 
69 
69 
69 
69 
69 
68 
69 
68 
68 
68 
68 
68 
68 
68 
67 
68 
67 

^ 
67 

67 

9.24  632 

74 
73 
74 
73 
74 
73 
73 
73 
73 
73 
72 

73 
72 

73 

72 
72 
72 
72 
72 
71 
72 

71 
72 
7i 
7i 
71 
71 
70 
7i 
7i 
70 

7° 
71 
70 

70 
70 
70 
69 

7° 
69 

7° 

g 

69 

69 
69 
69 
69 

69 
68 
69 
68 
68 
68 
68 
67 
68 
68 
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0.75  368 

9-99  335 

2 
2 
3 
2 
2 
2 

3 

2 
2 
2 

3 
2 
2 
2 

3 
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3 
2 
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2 
2 

3 
2 

3 

2 
2 

3 

2 

3 

2 

3 

2 
2 

3 

2 

60 

59 
58 
57 
56 
55 
54 
53 
52 
51 
50 

49 
48 

47 
46 
45 
44 
43 
42 
41 
40 

37 
36 
35 
34 
33 
32 
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30 
29 
28 
27 
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74     73      72 

7-4    7-3     7-2 
14.8  14.6  14.4 

22.2    21.9    21.6 
29.6    29.2    28.8 

37-o  36-5  36-0 
44.4  43.8  43.2 
51.8  51.1  50.4 
59-2  58-4  57-6 
66.6  65.7  64.8 

71      70      69 

7.1     7.0    6.9 
14.2  14.0  13.8 

21.3    21.0    20.7 
28.4    28.O    27.6 

35-5  35-°  34-5 
42.6  42.0  41.4 
49.7  49.0  48.3 
56.8  56.0  55.2 
63.9  63.0  62.1 

68     67      66 

6.8     6.7     6.6 
13.6  13.4  13.2 
20.4  20.  i   19.8 
27.2  26.8  26.4 
34-o  33.5  33.0 
40.8  40.2  39.6 
47.6  46.9  46.2 
54.4  53.6  52.8 
61.2  60.3  59.4 

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9.25000 
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0.75  294 
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0.75  074 
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0.74  781 
0.74  708 

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9.99  328 

9-99  326 
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9.24  677 

9-25  365 

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0-74  345 
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0.74  129 
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9-99  257 
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0.72  434 
0.72  365 
0.72  296 
0.72227 
0.72  158 
0.72089 
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0.71  951 
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9-99  241 
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9-99  219 

9.27471 

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L.  Cos.   |  d. 

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L.  Sin.       d. 

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0.68  194 

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9.32  810 
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44-1  43-4  42.7 

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9-33426 
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9-33  548 
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61 
61 
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0.66  574 
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0.66  391 
0.66  330 
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0.66  147 
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3 
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49 
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0.66  026 

9.98  986 

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22 
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24 
25 
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9.33018 
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9-33  133 
9-33  190 
9-33  248 
9-33  305 
9-33  362 
9-33  420 
9-33477 

57 

58 

57 
58 
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9-34  034 
9-34095 
9-34  155 
9-342I5 
9.34276 
9-34  336 
9-34  396 
9-344S6 
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61 
60 
60 
61 
60 
60 
60 
60 
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0.65  966 
0.65  905 
0.65  845 
0.65  785 
0.65  724 
0.65  664 
0.65  604 
0.65  544 
0.65  484 

9.98  983 
9.98  980 
9.98  978 

9-98  975 
9.98972 
9.98  969 
9.98  967 
9.98  964 
9.98961 

3 

2 

3 
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37 
36 
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31 
32 
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37 
38 
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9-33  647 
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9-33  93i 
9-33  987 
9-34  043 

56 

57 
57 
57 
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C-7 

9-34  635 
9-34  695 
9-34  755 
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9-34  874 
9-34  933 
9-34  992 
9-35051 
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60 
60 
59 
60 
59 
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eg 

0.65  365 
0.65  305 
0.65  245 
0.65  1  86 
0.65  126 
0.65  067 
0.65  008 
0.64  949 
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9-98955 
9-98  953 
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9.98  947 
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46 
46 

45 
46 

46 
46 

45 
46 

45 
46 

45 
46 

45 
45 
46 

45 
45 
45 
45 
45 
45 
45 
44 
45 
45 
45 
44 
45 
44 
45 
44 
45 
44 
44 
44 
45 
44 
44 

9.42  805 

51 
5° 
51 

50 
50 
5i 
50 
5° 
5° 
50 
50 
5° 
50 
50 
5° 
49 
5° 
50 
49 
5° 
49 
5° 
49 
5° 
49 
49 
49 
5° 
49 
49 
49 
49 
49 
49 
49 
48 
49 
49 
48 
49 
49 
48 
49 
48 
48 
49 
48 
48 
48 
49 
48 
48 
48 
48 
48 
48 
47 
48 
48 
48 

°-57  '93 

9.98  494 

3 
3 
4 
3 
4 
3 
3 
4 
3 
4 
3 
4 
3 
3 
4 
3 
4 
3 
4 
3 
4 
3 
4 
3 
3 
4 
3 
4 

3 

A 

3 
4 
3 
4 
4 
3 
4 
3 
4 
3 
4 
3 
4 
3 
4 
4 
3 
4 
3 
4 
3 
4 
4 
3 
4 
3 
4 
4 
3 
4 

60 

P 

57 
56 
55 
54 
53 
52 
5i 
5O 
49 
48 
47 
46 
45 
44 
43 
42 
41 
4O 
39 
38 
37 
36 
35 
34 
33 
32 
3i 
30 

29 
28 
27 
26 
25 
24 
23 

22 
21 
2O 
19 

18 
17 
16 
15 
H 
13 

12 
II 

10 

9 

8 

7 
6 
5 
4 
3 

2 
I 

0 

2      I 

3     i 
4    2 

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\\ 

9    A 

2 

3    i 

4     i 

3  \ 

7    3 
8    3 
9    4 

i 
2 

3 
4 
5 
6 

9 

2 

3 
4 

5 
6 

I 

9 

51      50     49 
5.1     5.0    4.9 

O.2    IO.O      9.8 

5-3  15-0  '4-7 

0.4    20.0    19.6 

5.5  25.0  24.5 

0.6  30.0  29.4 

5-7  35-o  34-3 
0.8  40.0  39.2 

5-9  45-o  44-i 

48     47      46 

4.8    4.7    4.6 
9.6    9.4    9.2 
4.4  14.1   13.8 
9.2  18.8  18.4 
40  23.5  23.0 
8.8  28.2  27.6 
3.6  32.9  32.2 
8.4  37.6  36.8 
3.2  42.3  41.4 

45       44 

4-5      44 
9.0      8.8 
13-5     13-2 
18.0     17.6 

22-5       22.0 
27.0       26.4 
31.5       30.8 
36.0       35.2 
40.5       39.6 

4         3 

0.4      0.3 
0.8      0.6 
1.2      0.9 

1.6          1.2 

2.0          1.5 

2.4       1.8 

2.8        2.1 

3-2      2.4 
3-6       2.7 

9-4i  347 
9-4i  394 
9.41  441 

9.41  488 

9-4i  535 
9.41  582 

9.41  628 
9.41  675 
9.41  722 

9-42  856 
9.42  906 
y-42957 
943  007 
943057 
943  108 

943  15s 
943  208 
943  258 

0.57  144 
0.57094 
0.57043 
0.56  993 
0.56  943 
0.56  892 
0.56  842 
0.56  792 
0.56  742 

9.98491 
9.98  488 
9.98  484 
9.98481 
998477 
9.98  474 
9.98471 
9.98  467 
9.98  464 

9.41  768 

9-43  308 

0.56  692 

9.98460 

9.41  815 
9.41  861 
9.41  908 

9-41  954 
9.42001 
9.42  047 
9.42  093 
9.42  140 
9.42  186 
942  232 
9.42  278 
9.42  324 
9.42  370 
9.42416 
9.42  461 
9-42  5°7 
9-42  553 
9.42  599 
9-42  644 

943  358 
9-43  408 
9-43  458 
943  5°8 
943  55» 
943  607 
9-43  657 
943  707 
943  756 

0.56  642 
0.56  592 
0.56  542 
0.56492 
0.56  442 
0.56393 
0-56  343 
0.56  293 
0.56  244 

9.98457 
9-98453 
9.98  450 

9.98  447 
9-98  443 
9.98  440 
9.98436 
9-98433 
9-98  429 

9.43  806 

0.56  194 

9.98  426 

943  855 
943  903 
9-43  954 
9.44004 

9-44053 
9.44  102 

9.44I5I 

9.44  201 

9-44  250 

0.56  143 
0.56095 
0.56  046 

0-55  996 
o-55  947 
0.55  898 

0.55  849 
o-55  799 
o-55  75° 

9.98  422 
9.98419 
9.98415 
9.98412 
9.98  409 
9.98  405 
9.98  402 
9.98  398 
9-98  393 

944  299 

o-55  701 

9.98391 

9-42  735 
9.42  781 
9.42  826 
9.42  872 
9.42917 
9.42  962 
9.43  008 

9-43  053 
9.43  098 

9-44  34« 
9-44  397 
9.44446 

944  493 
9-44  544 
944  592 
9.44641 
944  690 
9-44  738 

0-55  652 
o-55  603 
0-55  554 
o-55  5°5 
0.55  456 
0.55  408 

0-55  359 
0-55  3'o 
0.55  262 

9.98  388 
9.98  384 
9.98381 

9-98  377 
9-98  373 
9.98  370 
9.98  366 
9-98  363 
9-98  359 

9-43  143 

944  787  _ 
944  836 
9-44  884 
9-44  933 
9.44981 
945  029 
945  o?^ 
945  I26 
945  J74 
9-45  222 

o-55  213 
0.55  164 
°-55  n6 
0.55  067 

0-55  019 
0.54971 
0.54  922 
0.54  874 
0.54  826 
0-54  778 

9.98  356 
9.98352 
998349 
9-98  345 
9.98  342 
9.98  338 
9-98  334 

9-9833I 
9.98  327 
9.98  324 

9-43  1  88 
9-43  233 
9.43  278 

9-43  323 
9-43  367 
9.43412 

943457 
9-43  502 
9-43  546 

4444 

50     49     48     47 

o' 
6.2    6.1    6.0    5.9 

•  i  S.  Si  8.4  1  8.0  1  7.6 
,31.230.630.029.4 
•j  43.8  42-9  42-0  41.  i 

3333 
51     50    49    48 

°|  8.5    8.3    8.2    8.0 
,  25.5  25.0  24.5  24.0 
;  42.5  4  1.  7  40.8  40.0 

9-43591 

945  271 

0.54  729 

9.98  320 

9-43  635 
9-43  680 
9-43  724 
9-43  769 
9-438I3 
9-43  857 

9-43  901 
9.43  946 
9-43  990 

945  3'9 
945  367 
945413 
945  463 
9455H 
9-45  559 
945  606 
9-45  654 
945  702 

0.54681 
0-54  633 
0-54  585 

0-54  537 
0.54489 
0.54441 

0.54  394 
0.54  346 
0.54  298 

9-98317 
9-983I3 
9.98  309 

9.98  306 
9.98  302 
9-98  299 
9.98  295 
9.98  291 
9.98  288 

9-44  034 

945  73° 

0.54  250 

9.98  284 

L.  Cos.      d. 

L.  Cot.   lc.  d.    L,  Tan, 

L.  Sin.    |  d. 

1 

P.  P. 

74' 


436 


16° 


' 

L.  Sin. 

d. 

L.  Tan.    c.  d.     L.  Cot. 

L.  Cos. 

d. 

p.  p. 

0 

2 

3 

4 

I 

9 
IO 

n 

12 
13 
14 

17 

18 
19 
2O 

21 
22 

23 
24 
25 
26 

3 

29 
30 

3i 

32 
33 
34 

i 

39 
40 

42 
43 
44 
45 
46 

47 
48 
49 
50 

53 

54 

P 

59 
60 

9.44  034 

44 
44 
44 
44 
43 
44 
44 
44 
43 
44 
44 
43 
43 
44 
43 
44 
43 
43 
43 
43 
43 
44 
43 
42 

43 
43 
43 
43 
43 
42 

43 
42 
43 
42 
43 
42 

42 

42 

43 
42 
42 
42 
42 
42 
42 
42 
4i 
42 

42 
42 
4» 
42 
4i 
42 
4i 
42 
4i 
42 

9-45  73° 

47 
48 
47 
48 

3 

47 
48 

47 
47 
47 

47 
47 
47 
47 
47 
47 
47 
46 

47 
47 
47 
46 
47 
47 

46 

47 
46 
46 
47 
46 
46 
46 
46 
46 
46 

46 
46 
46 
46 
45 
46 

46 
46 

45 
46 

45 

S 

45 

8 

45 

o-54  250 

9.98  284 

3 
4 
4 

4 
4 
3 
4 
4 
3 
4 
4 
3 
4 
4 
3 
4 
4 
3 
4 
4 
3 
4 
4 
4 
3 
4 
4 

3 
4 

4 
4 
3 

4 
3 
4 
4 
4 
3 
4 
4 
4 
4 
3 
4 
4 
4 
4 
4 
3 
4 
4 
4 
4 
4 
4 
3 

60 

59 
58 
57 
56 
55 
54 
53 
52 
5i 
50 

49 
48 

47 
46 
45 
44 

43 
42 

40 

38 
37 
36 
35 
34 

32 
30 

27 
26 
25 
24 

22 
21 
2O 

19 

18 

16 
15 

14 
13 

12 

IO 

7 
6 
5 
4 
3 

2 
I 

o 

2 

3    i 

4     i 
5    2 

6      2 

11 

9    A 
i 

2 

3     i 
4     i 

1    \ 

8    2 
9    4 

i 

2 

3 
4 

I 

I 

9 

i 

i 

i 

i 

9 

48     47      46 

4.8    4.7     4.6 
9-6     9-4     9-2 
4.4  14.1   13.8 
9.2  18.8  18.4 
4.0  23.5  23.0 
8.8  28.2  27.6 
3.6  32.9  32.2 
8.4  37.6  36.8 
3.2  42.3  41.4 

45     44      43 

4-5     4-4     4-3 
9.0     8.8     8.6 
3.5   13.2  12.9 
8.0  17.6  17.2 

2-5    22.0   21.5 
7-0    26.4   25.8 
1.5    30.8   30.1 

6.0  35.2  34.4 
o-5  39-6  38-7 

42      41 

4.2      4.1 

8.4        8.2 

12.6    12.3 
16.8    16.4 
21.0    20.5 
25.2    24.6 
29.4    28.7 
33-6    32-8 
37-8    36-9 

4          3 

0.4        0.3 
0.8        0.6 
1.2        0.9 

1.6            1.2 
2.0            1.5 

2.4        1.8 

2.8            2.1 

3-2        2.4 
3-6        2.7 

9.44078 

9.44  122 

9.44  1  66 

944  210 

9-44  253 
9-44  297 
9-44  341 
9-44  385 
9-44  428 

9-45  797 
9-45  845 
9-45  892 
9.45  940 

9-45  987 
9.46  035 

9.46  082 
9.46  130 
9-46  177 
9.46  224 

0.54  203 

o-54  155 
0.54  108 

0.54060 
0.54013 
0-53  965 
0.53918 
0.53  870 
0-53823 

9.98  281 
9.98  277 
9-98  273 
9.98  270 
9.98  266 
9.98  262 
9.98  259 
9-98  255 
9.98251 

9.44472 

0.53  776 

9.98  248 

9.44516 
9-44  559 
9.44  602 

9.44  646 
9-44  689 
9-44  733 

9-44  776 
9.44819 
9.44  862 

9.46  271 
9-463I9 
9.46  366 

9.46413 
9.46  460 
9.46  507 

946  554 
9.46  60  1 
9.46  648 

0.53  729 
0.53681 
0-53  634 
o-53  587 
o-53  540 
o-53  493 
0.53  446 
o-53  399 
0.53352 

9-98  244 
9.98  240 
9.98  237 
9-98  233 
9.98  229 
9.98  226 

9.98  222 
9.98218 
9.98215 

9-44  905 

9-46  694 

0.53  306 

9.98  211 

9.44  948 
9-44  992 
9-45  °35 

9-45  077 
9.45  120 

9-45  163 
9  45  206 
9-45  249 
9-45  292 

9.46  741 
9.46  788 
9.46  835 
9.46881 
9.46928 
9.46  975 
9.47021 
9.47  068 
9-47  "4 

0-53  259 
0.53212 

o-53  165 
0.53119 
0.53072 
0-53025 
0.52979 
0.52932 
0.52886 

9.98  207 
9.98  2O4 
9.98  200 
9.98  196 
9.98  192 
9.98  189 
9-98  185 

9.98  181 
9.98177 

9-45  334 

9.47  1  60 

0.52  840 

9.98  174 

9-45  377 
9-45  4i9 
9-45  462 
9-45  5°4 
9-45  547 
9-45  589 
9-45  632 
9-45  674 
9-45  7i6 
9-45  758 
9.45  801 
9-45  843 
9-45  885 
9-45  927 
9-45  969 
9.46011 

9.46053 
9.46095 
9.46  136 

9-47  207 
9-47  253 
9-47  299 
9-47  346 
9-47  392 
9-47  438 
9-47  484 
9-47  530 
9-47  576 

0-52  793 
0-52  747 
0.52  701 

0.52654 
0.52608 
0.52  562 
0.52  516 
0.52  470 
0.52424 

9.98170 
9.98  1  66 
9.98  162 
9.98  159 
9.98  155 
9.98151 
9.98  147 
9.98  144 
9.98  140 

9.47  622 

0.52378 

9.98  136 

9.47  668 
9.47  7'4 
9-47  76o 
9.47  806 
947852 
9-47  897 
9-47  943 
9-47  989 
9.48  035 

0.52  332 
0.52  286 
0.52  240 
0.52  194 
0.52  148 
0.52  103 
0.52057 
0.52011 
0.51965 

9-98  132 
9.98  129 
9.98125 

9.98  121 
9.98117 
9.98lI3 

9.98  no 
9.98  106 
9.98  102 

4444 
48    47    46     45 

0    6.0    5.9    5.8    5.6 
1  8.0  17.6  17.2  16.9 
2  30.0  29.4  28.8  28.1 
3  42.0  41.  i  40.2  39.4 

3333 
48    47     46    45 

0    8.0    7.8    7.7    7.5 
24.023.523.022.5 
3  40.0  39.2  38.3  37.5 

9.46178 

9.48  080 

0.51  920 

9.98098 

9.46  220 
9.46  262 
9-46  303 

946  386 
9.46  428 
9.46469 
9.46511 
9-46552 

9.48  126 
9.48171 
9.48217 
9.48  262 
9.48  307 
948  353 
9-48  398 
9-48  443 
9.48  489 

0.51874 
0.51  829 
0.51  783 

0-51  738 
0.51  693 
0.51  647 
0.51  602 

o-S1  557 
0.51511 

9.98  094 
9.98  090 
9.98087 
9.98  083 
9.98  079 
9.98  075 
9.98071 
9.98  067 
9.98063 

9-46  594 

9-48  534 

0.51  466 

9.98  060 

L.  Cos. 

d. 

L.  Cot.    c.  d.i    L.  Tan. 

L.  Sin.    I  d. 

' 

P.P. 

73° 


17° 


437 


1 

L.  Sin. 

d. 

L.  Tan. 

c.  d. 

L.  Cot. 

L.  Cos. 

d. 

P.  P. 

o 

9.46  594 

41 

948  534 

41: 

0.51  466 

9.98  060 

4 

60 

I 

2 

3 
4 

I 

7 
8 

9 

946  635 
9.46  676 
9.46717 
9.46  758 
9.46  800 
9.46841 
9.46  882 
9.46  923 
9.46  964 

4i 
4i 
4i 

42 
4i 
4i 
4i 
4i 

948  579 
9.48  624 
9.48  669 
9.48  714 
9-48  759 
9.48  804 

9.48849 
9.48  894 
9-48  939 

45 
45 
45 
45 
45 
45 
45 
45 

0.51421 
0.51  376 
0-51  33i 
0.51  286 
0.51  241 
0.51  196 

0.51  151 
0.51  106 
0.51  06  1 

9.98  056 
9.98052 
9.98  048 
9.98  044 
9.98  040 
9.98  036 
9.98  032 
9.98  029 
9.98  025 

4 
4 
4 
4 
4 
4 
3 
4 

59 
58 
57 
56 
55 
54 
53 
52 
51 

2 

3 

4 

5 
6 

7 

45     44     43 

4-5     4-4    4-3 
9.0    8.8     8.6 
13.5  13.2  12.9 
18.0  17.6  17.2 

22-5    22.O   21.5 
27.0    26.4   25.8 
31.5    3O.8    30.1 

IO 

9-47  oo5 

9.48  984 

0.51  016 

9.98021 

5O 

8 

36.0   35.2    34.4 

12 
13 

14 
15 
16 

19 

9-47  °45 
9.47  086 
9.47  127 
9.47  1  68 
9-47  209 
9.47  249 

9-47  29° 
9-47  330 
9-47  37i 

4i 
4i 
4i 
4i 
40 

4i 
40 

4i 
40 

9.49  029 
9-49  073 
9.49  118 
9.49  163 
9.49  207 
9.49  252 
9-49  296 
9-49  341 
9-49  385 

44 
45 
45 
44 
45 
44 
45 
44 
41: 

0.50971 
0.50927 
0.50  882 
0.50  837 

°-5°  793 
0.50  748 

0.50  704 
0.50  659 
0.50615 

9.98017 
9.98013 
9.98  009 
9.98005 
9.98001 
9-97  997 
9-97  993 
9-97  989 
9-97  986 

4 
4 
4 
4 
4 
4 
4 
3 

49 
48 
47 
46 
45 
44 

43 

42 
4i 

9 

i 

2 

3 
4 

40-5  39-6  38-7 

42     41      40 

4.2    4.1     4.0 
8.4    8.2    8.0 

12.6    12-3    I2.O 

1  6.8  16.4  16.0 

20 

9.47411 

41 

9-49  430 

0.50570 

9.97  982 

4O 

g 

25.2  24.6  24.0 

21 
22 
23 
24 
25 
26 

27 
28 
29 

9.47452 
9.47  492 
9-47  533 

9-47  573 
9.47613 

9-47  654 
9.47  694 
9-47  734 
9-47  774 

40 
4i 
40 
40 
4i 
40 
40 
40 

9.49  474 
9-495I9 
9-49  563 

9-49  6°7 
9.49  652 
9.49  696 

9-49  740 
9-49  784 
9.49  828 

45 
44 

44 
45 
44 
44 
44 
44 

0.50526 
0.50481 
0.50  437 

0-50  393 
0.50  348 
0.50  304 
0.50  260 
0.50  216 

0.^0  1/2 

9.97978 

9-97  974 
9.97  970 

9.97  966 
9.97  962 
9-97  95s 
9-97  954 
9-9795° 
9-97  946 

4 
4 

4 
4 
4 
4 
4 
4 

39 
38 
37 
36 
35 
34 
33 
32 
3i 

9 
i 

2 

29.4  28.7  28.0 
33.6  32.8  32.0 
37-8  36-9  36-0 

39    5      4     3 

3.9  0.5  0.4  0.3 
7.8  i.o  0.8  0.6 

3O 

9.47814 

9.49  872 

0.50  128 

9-97  942 

3O 

3 

11.7  1.5  1.2  0.9 

3i 

32 
33 
34 

P 
P 

9-47  854 
9.47  894 

9-47  934 

9-47  974 
9.48014 
9.48  054 
9.48  094 
948  133 

40 
40 
40 
40 
40 
40 
39 

9.49916 
9.49  960 
9.50  004 
9.50  048 
9.50092 
9.50  136 
9.50  1  80 
9.50  223 

44 

44 
44 
44 
44 
44 
43 

0.50  084 
0.50  040 
0.49  996 
0.49  952 
0.49908 
0.49  864 
0.49  82O 

o-49  777 

9-97  938 
9-97  934 
9-97  930 
9.97  926 
9.97922 
9.97918 
9.97914 
9.97910 

4 
4 
4 
4 
4 
4 
4 

29 
28 

27 
26 

25 
24 

23 

22 

4 

I 

7 
S 

9 

15.6   2.O    1.6    1.2 
19.5    2.5    2.O    1.5 

23.4  3.0  2.4  1.8 

27.3    3.5    2.8    2.1 

31.2  4.0  3.2  2.4 
35.1  4.5  3.6  2.7 

39 

9-48I73 

4° 

9.50  267 

44 

0-49  733 

9-97  9o6 

4 

21 

40 

9.48213 

9.50311 

o.4y  689 

9.97  902 

20 

4i 
42 
43 
44 
45 
46 

S 

49 

9.48  252 
9.48  292 
948  332 
9.48371 
9.48411 
9.48  450 
9.48  490 
9.48  529 
9.48  568 

40 
40 

39 
40 
39 
40 

39 
39 

9-5°  355 
9.50398 
9.50  442 
9.50  485 
9-5°  529 
9-50572 
9.50616 
9-5°659 
9-50703 

43 
44 
43 
44 
43 
44 
43 
44 

0.49  645 
0.49  602 
0.49  558 

0-49  5  »5 
0.49471 
0.49  428 
0.49  384 
0.49341 
0.49  297 

9-97  898 
9-97  894 
9-97  890 
9.97  886 
9.97  882 
9.97878 

9-97  874 
9.97  870 
9.97  866 

4 
4 
4 
4 

4 
4 
4 
4 

19 
18 

'7 
16 
15 
14 
'3 

12 
II 

0 
2 

3 
4 
5 

544 
43     45      44 

4-3     5-6    5-5 
12.9  16.9  16.5 
21.5  28.1  27.5 
30.1  39.4  38.5 
38-7    -     - 

50 

9.48  607 

9.50  746 

0.49  254 

9.97  8(»i 

IO 

5i 

52 
53 
54 

P 
!i 

59 

9.48  647 
9.48  686 
9.48  725 
9.48  764 
9.48  803 
9.48  842 
9.48  881 
9.48  920 
9.48  959 

39 
39 
39 
39 
39 
39 
39 
39 

^Q 

9.50789 

9-5°  833 
9.50876 

9.50919 
9.50  962 
9-5  l  °°5 
9.51  048 
9.51092 
9-Si  '35 

44 
43 
43 
43 
43 
43 
44 
43 

0.49211 
0.49  167 
0.49  124 
0.49081 
0.49  038 
0.48  995 
0.48952 
0.48  908 
048865 

9-97  857 
9-97  853 
9.97  849 

9-97  845 
9-97  841 
9-97  837 
9-97  833 
9-97  829 
9.97  823 

4 
4 
4 
4 
4 
4 
4 
4 

9 
8 
7 
6 
5 
4 
3 

2 

I 

o 
i 

2 

3 
4 

433 
43     45      44 

5-4    7-5     7-3 

I6.I     22-5    22.0 

26.9  37-5  36-7 
37-6    —     — 

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9.48  998 

9.51  178 

0.48  822 

9.97  821 

O 

L.  Cos. 

d. 

L.  Cot. 

c.  d. 

L.  Tan. 

L.  Sin. 

d. 

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P.P. 

72° 


438 


18° 


' 

L.  Sin. 

d.      L.  Tan. 

c.  d.     L.  Cot.       L.  Cos.      d. 

p.p. 

0 

9.48  998 

9.51  178 

0.48822 

9.97  821 

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9-49  037 

9.51  221 

0.48  779 

9.97817 

59 

2 

9-49  076 

39 

9-5  1  264 

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0.48  736 

9.97812 

5 

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3 

949"5 

39 

9.51  306 

42 

0.48  694 

9.97  808 

4 

S7 

43     42     41 

4 
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9-49  153 
9-49  192 

^ 
39 

9-51  349 
9-5  !  392 

43 
43 

0.48651 
0.48  608 

9-97  804 
9.97  800 

4 
4 

56 

55 

I 

2 

4-3     4-2     4-1 
8.6    8.4    8.2 

6 

9-49  231 

39 

9-5M35 

43 

0.48  565 

9-97  796 

54 

3 

12.9  12.6  12.3 

7 
8 

9 
10 

9-49  269 
9-49  308 
9-49  347 

3s 
39 
39 
38 

9-5I478 
9-51  520 
9-51  563 
9.51  606 

43 
42 
43 

43 

0.48  522 
0.48  480 
0.48  437 

9-97  792 
9-97  788 
9-97  784 

4 
4 
4 
5 

53 

S2 

50 

4 

i 

i 

17.2  16.8  16.4 
21.5  21.0  20.5 
25.8  25.2  24.6 
30.1  29.4  28.7 
34.4  33.6  32.8 

9-49  385 

0.48  394 

9.97  779 

n 

12 

9-49  424 
9-49  462 

38 
78 

9.51  648 
9.51  691 

43 

0.48  352 
0.48  309 

9-97  775 
9.97771 

4 

49 
48 

9 

38.7  37.8  36.9 

13 

9-49  5°° 

9-5  !  734 

43 

0.48  266 

9-97  767 

47 

I4 

9-49  539 

39 

78 

9.51  776 

42 

0.48  224 

9-97  763 

46 

IS 

9-49  577 

78 

9.51819 

0.48  181 

9-97  759      7 

45 

39     38      37 

16 

9.49615 

9.51  861 

0.48  139 

9-97  754 

3 

44 

i 

3-9     3-8     3.7 

17 
18 

9-49  654 
9-49  692 

38 
78 

9.51903 
9.51946 

43 

0.48097 
0.48  054 

9-97  75° 
9-97  746 

4 

43 

42 

2 

S 

7-8     7-6     7-4 
11.7  11.4  n.  i 

19 

9-49  730 

9.51  988 

0.48012 

9.97  742 

4 

15.6  15.2  14.8 

20 

9-49  7<>8 

38 

9.52031 

42 

o-47  969 

9-97  738 

40 

5 

19.5   19.0  18.5 

21 

9.49  806 

78 

9.52073 

0-47  927 

9-97  734 

39 

0 

23.4    22.8    22.2 

22 
23 

9-49  844 
9.49  882 

38 

9-52115 
9-52  157 

42 

0.47  883 
0-47  843 

9-97  729 
9-97  725 

5 
4 

38 
37 

7 
8 

27.3    26.6    25.9 
31.2   30.4    29.6 

24 

9-49  920 

^ 

78 

9.52  200 

43 

0.47  800 

9-97  72i 

76 

y 

35-1  34-2  33-3 

III 

9-49  95s 
9-49  996 

38 

9.52  242 
9.52  284 

42 

o-47  758 
0.47  716 

9.97717 
9.977U 

4 

35 
34 

11 

29 

9-5°  °34 

9.50072 
9.50  no 

35 

38 
38 

9-52  326 
9-52  368 
9.52410 

42 
42 
42 

42 

o-47  674 
o-47  632 
0-47  590 

9-97  7°8 
9-97  7°4 
9-97  700 

b 
4 
4 

33 

32 

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36       5       4 

3.6    0.5    0.4 

30 

9.50  148 

9-52452 

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9.97696  i    7 

30 

31 
32 
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9-5°  l85 
9.50223 
9.50  261 

38 
38 

9-52  494 
9-52  536 
9.52  578 

42 
42 

0.47  506 

0-47  464 
0.47  422 

9.97691 
9-97  687 
9-97  683 

3 

4 
4 

29 
28 
27 

4 
5 
6 

14.4    2.0    1.6 
1  8.0    2.5    2.0 
21.6    3.0    2.4 

;   14 

9.50  298 

78 

9.52  620 

0.47  380 

9-97  679 

26 

25.2    3.5    2.8 

35 

9-5°  336 

78 

9.52  661 

0-47  339 

9-97  674 

25 

8 

28.8  4.0  3.2 

36 

9-5°  374 

9.52  703 

0.47  297 

9-97  67° 

24 

9 

32.4  4.5  3.6 

i  ^7 

9.50411 

78 

9-52  745 

0-47  255 

9.97  666 

23 

38 

9-50449 

77 

9-52  787 

o-47  213 

9.97662  i    * 

22 

39 

9.50486 

9.52  829 

41 

0-47  I71 

9-97657  i    i 

21 

4O 

9-50523 

7,8 

9.52870 

42 

0.47  130 

9-97653  ! 

20 

41 

9.50561 

9.52912 

0.47  088 

9.97649  i 

19 

555 

42 

9-5°  598 

37 

9-52953 

42 

0.47  047 

9-97645  i    e 

18 

43      42      41 

43 

9.50  635 

->8 

9-52995 

o-47  °°5 

9.97640       3 

17 

44 

9.50  673 
9.50710 
9-5°  747 

J5 
37 
37 

9-53037 
9-53078 
9-53  120 

42 

42 

0.46  963 
0.46  922 
0.46  880 

9-97  636 
9-97  632 
9.97  628 

16 
15 

i 

2 

3 

4-3     4-2    4-i 
12.9  12.6  12.3 

21.5    2I.O    20.5 

47 

;  48 

9.50  784 
9.50821 

37 

37 

9-53  161 
9-53  202 

41 

4i 

0.46  839 
0.46  798 

9-97  623 
9.97  619 

} 

13 

12 

4 
5 

30.1    29.4   28.7 

38-7  37-8  36-9 

49 

9.50  858 

18 

9-53  244 

0.46  756 

9.97613 

II 

5O 

9.50  896 

9-53  285 

0.46715 

9.97610 

10 

51 

9-5°  933 

9-53  327 

0.46  673 

9.97606  ;    ; 

9 

44.         4 

52 

9-5°  97° 

9.53368 

0.46  632 

9.97  602 

8 

5S 

9.51  007 

9-53  409 

41 

0.46  591 

9-97  597 

5 

7 

43     42     41 

54 

SS 

9.51043 
9.5  1  080 

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37 

9-5345° 
9-53492 

42 

0.46  550 
0.46  508 

9-97  593 
9-97  589 

4 
4 

6 

S 

o 

i 

5-4    5-2     5-1 
16.1   15.8  15.4 

S6 

9-51  "7 

9-53533 

41 

0.46  467 

9-97  584 

4 

~ 

26.9  26.2  25.6 

9-5  1  J54 

37 

9-53  574 

0.46  426 

9-97  580 

T, 

3 

37.6  36.8  35.9 

58 

9.51  191 

37 
76 

9-536I5 

0-46  383 

9-97  576 

2 

S9 

9.51  227 

77 

9-53656 

41 

0.46  344 

9-97  57  1 

I 

60 

9.51  264 

9-53697 

0.46  303 

9-97  567 

0 

L.  Cos. 

d. 

L.  Cot.    |c.  d.    L.  Tan. 

L.  Sin. 

d.  |    ' 

P.P. 

71° 


19° 


439 


/ 

L.  Sin. 

d. 

L.  Tan.    c.  d.     L.  Cot. 

L.  Cos.       d. 

p.  p. 

o 

9.51  264 

9-53697      ,, 

0.46  303 

9-97567  '    „ 

60 

I 

9.51  301 

9-53  738 

0.46  262 

9-97  563 

S9 

2 

3 

9oi  338 
9-51  374 

37 
36 

9-53  779 
9.53  820 

41 

0.46  221 
0.46  1  80 

9-97  558 
9-97  554 

5 
4 

58 
57 

41      40      39 

4 
5 

9.51411 

37 
36 

9.53861 
9-53  9°2 

41 
41 

0.46  139 
0.46  098 

9-97  55° 
9-97  545 

5 

56 

SS 

2 

4.1     4.0     3.9 
8.2     8.0     7.8 

6 
9 

9.51  484 
9.51  520 
9.51557 
9-51  593 

37 
36 
37 

9-53  943 

9-53  984 
9-54025 
9.54065 

41 

4i 
4i 
40 

0.46  057 
0.46016 

0-45  975 
0-45  935 

9-97  54i 
9-97  536 
9-97  532 
9-97  528 

5 
4 
4 

54 
53 

52 

3 
4 

i 

12-3    I2.O    II-7 

16.4  16.0  15.6 

20.5    20.0    19.5 
24.6   24.0   23.4 

IO 

9.51  629 

37 

9-54  106   ;, 

0.45  894 

9-97  523 

50 

8 

32.8    32.0   31.2 

ii 

9.51  666 

9-54  H7 

0-45  853 

9-97  5f9 

49 

9 

36.9    36.0    35.1 

12 

9.51  702 

36 

9-54  187 

o-45  813 

9-97  5'5 

48 

13 

9-51  738 

9-54  228 

o-45  772 

9.97510 

47 

14 

IS 

9-51  774 
9.51811 

37 

9-54  269 
9-54  309 

40 

0-45  73i 
0.45  691 

9-97  5°6 
9-97  501 

5 

46 
4S 

37      36      35 

16 

9-51  847 

9-54  35° 

0.45  650 

9-97497 

44 

I 

3-7     3-6     3-5 

17 
18 

19 
20 

9.51  883 
9-5I9I9 
9-51  955 

I 

36 

9-54  390 
9-54431 
9-54471 

40 
4* 
40 

0.45  610 
0.45  569 
o-45  529 

9-97492 
9-97488 
9-97  484 

4 
4 
5 

43 
42 

40 

2 

3 
4 

5 

5 

7-4     7-2     7-° 
1  1.1   10.8  10.5 
14.8  14.4  14.0 
18.5  18.0  17.5 

9J>£99J 

9.54512 

0.45  488 

9-97  479 

21 

22 
23 

9.52027 
9.52063 
9.52099 

36 

36 

9-54552 
9-54  593 
9-54  633 

41 
40 
40 

0-45  448 
0.45  407 
0.45  367 

9-97475      7 
9.97470 
9-97  466      J 

39 
38 
37 

7 
8 
9 

25.9  25.2  24.5 
29.6  28  8  28.0 
33-3  32.4  *i.c 

24 

9-52  135 

j6 

9-54673 

41 

o-45  327 

9.97461 

36 

25 

9.52171 

36 

9-54  7'4 

40 

0.45  286 

9-97457 

35 

26 

9.52  207 

9-54  754 

0.45  246 

9-97  453 

34 

27 

9.52  242 

35 

9-54  794 

0.45  206 

9-97  448 

33 

34      5      4 

28 

9-52278 

36 

9-54835 

40 

0.45  165 

9-97  444      J 

32 

i 

3.4    0.5    0.4 

29 

9-52  3J4 

9.54  8/5 

AO 

0-45  125 

9-97  439       ;| 

31 

2 

6.8    i.o    0.8 

30 

9-52  35° 

9.54915     ^ 

0.45  085 

9-97435       P 

30 

3 

10.2    1.5     1.2 

32 

9-52  385 
9.52421 

36 

9-54  955     4o 
9-54995     Tn 

0.45  045 
0.45  005 

9-9743°  i 
9-97  426      7 

29 
28 

4 

13.6    2.0    1.6 

17.0      2.5      2.0 

33 
34 

P 

9-52456 
9.52492 
9-52527 
9-52  563 

36 

35 
36 

9-55035 
9.55075 
9-55  "5 
9-55  155 

40 
4O 

4° 

0.44  965 
0.44  925 
0.44  885 
0.44  845 

9.97421 
9.97417 
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9.97  408 

4 
5 
4 

27 
26 
25 
24 

7 
8 

9 

20.4      3-O      2.4 

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27.2    4.0    3.2 
30.6    4.5    3.6 

37 

9-52598 

35 

9-55  195 

0.44  805 

9-97  403 

5 

23 

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38 

9.52  634 

3*-* 

9-55  235 

0.44  765 

9-97  399 

22 

39 

9.52669 

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40 

0.44  725 

9-97  394      ;| 

21 

4O 

9.52  705 

9-553I5 

40 

0.44  685 

9-97390       r 

20 

41 

9.52  740 

9-55  355 

0.44  645 

9.97385  | 

19 

555 

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9-52  775 

9-55  395  !  "g 

0.44  605 

9.97381       * 

18 

41      40      39 

43 

9.52  811 

9-55  434     ~ 

0.44  566 

9-97  376       5 

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44 
45 
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9.52  846 
9.52881 
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35 
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9-55474 
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40 
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9-97  372 
9-97  367 
9-97  363 

5 
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16 
15 
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12-3    12.0    II-7 
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28.7    28.0    27.3 

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9-55  593     "o 

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36.9    36.0    35.1 

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9-55  633     40 

0.4-4  367 

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0.44327 

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II 

50 

9-53056 

y.;^  712       An 

0.44  288 

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10 

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9-53  092 

9.55  752 

39 

0.44  248 

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52 
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9.53126 
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0.44  209 
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8 
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41      40      39 

54 

9-53  196 

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9-55  870 

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0.44  130 

9-97  326 

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9-55  910 

0.44  090 

9-97  322 

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15.4    15.0    14.6 

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9-53  266 

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25.6   25.0    24.4 

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35-9  35-°  34-1 

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9-97  303 

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9-53  405 

9.56  107 

1  0-43893 

9-97  299 

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L.  Cos.      d. 

L.  Cot.    c.  d.    L.  Tan. 

L.  Sin.    |  d.      ' 

P.P. 

70' 


440 


20' 


L.  Sin. 

d. 

L.  Tan. 

c.  d. 

L.  Cot. 

L.  Cos. 

d. 

p.p. 

0 

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2 

3 
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12 
13 
14 

17 
18 

19 
20 

21 

22 
23 
24 

25 
26 

3 

29 
30 

31 
32 
33 

34 

3 

37 

39 
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42 
43 
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47 
48 

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52 
53 
54 
55 
56 
57 
58 
59 
60 

9-53  405 

35 
35 
34 

34 
35 
34 
35 
34 
35 
34 
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34 
34 
35 
34 
34 
34 
34 
34 
34 
34 
34 
34 
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34 
34 

9-56  107 

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39 
40 

39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
38 
39 
39 
39 
38 
39 
39 
38 
39 
38 
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39 
38 
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0-43  893 

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5 
4 
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5 
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5 
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4 
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5 
5 
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5 
5 
4 
5 
5 
5 
4 
5 
5 
4 

r 

4 
5 
5 
5 
4 
5 
5 
5 
5 
4 

5 
5 
5 
4 
5 
5 
5 
5 
4 
5 
5 
5 
5 

60 

H 

57 
56 
55 
54 
53 
52 
51 
50 

49 
48 

47 
46 
45 
44 
43 
42 
4i 
40 

3 

37 
36 
35 

34 
33 

3i 
30 

29 

28 

27 
26 
25 
~4 
23 

22 
21 
2O 

19 

18 

17 
16 
15 
H 
'3 

12 
II 

IO 

9 
8 

7 
6 
5 
4 
3 

2 

I 

0 

2 

3 
4 

5 
6 

I 

9 

2 

3 

4 

i 

i 

9 
i 

2 

3 
4 

I 

9 

40     39      38 

4-0     3-9     3-8 
8.0     7.8     7.6 
12.0  11.7  11.4 
16.0  15.6  15.2 
20.0  19.5  19.0 

24-0    23.4    22.8 
28.O    27.3    26.6 
32.0    31.2    30.4 
36.0    35.1    34.2 

37      35      34 

3-7     3-5     3-4 
7.4     7.0    6.8 
1  1.  1   10.5  10.2 
14.8  14.0  13.6 
18.5  17.5   17.0 

22.2    21.0    20-4 
25.9    24.5    23.8 
29.6    28.0    27.2 

33-3  31-5  30.6 

33       5        4 

3.3     0.5     0.4 
6.6     i.o    0.8 

9-9     i-5     1-2 
13.2     2.0     1.6 

16.5       2.5       2.0 
19.8       3.0       2.4 

23.1     3.5     2.8 
26.4    4.0    3.2 
29.7   4.5    3.6 

9-53  440 
9-53475 
9-53  5°9 
9-53  544 
9-53578 
9-53613 
9-53  647 
9-53  682 
9-537I6 

9.56  146 
9.56  185 
9.56  224 
9.56  264 
9-56  303 
9.56342 

9.56  381 
9.56  420 
9-56459 
9.56  498 

9-56  537 
9-56576 
9-56615 
9.56  654 
9-56  693 
9-56  732 
9.56771 
9.56  810 
9.56  849 

0.43  854 
0.43  815 
0.43  776 

0-43  736 
0.43  697 
0.43  658 
0.43  619 
0.43  580 
0.43  541 

9.97  294 
9-97  289 
9-97  285 
9.97  280 
9-97  276 
9.97271 
9.97  266 
9.97  262 
9-97  257 

9-53  75i 

0.43  502 

9.97252 

9-53  785 
9.53819 

9-53  854 
9.53888 
9-53922 
9-53957 

9-53991 
9.54025 
9-54059 

0-43  463 
0.43  424 

0-43  385 
0-43  346 
0-43  307 
0.43  268 

0.43  229 
0.43  190 
0.43151 

9-97  248 
9-97  243 
9-97  238 
9-97  234 
9-97  229 
9-97  224 
9.97  220 
9.97215 
9.97  210 

9-54  093 

9.56  887 
9.56  926 
9-56  965 
9.57004 

9.57042 
9.57081 
9-57  120 
9-57  »58 
9-57  r97 
9-57  235 

0.43113 

9.97  206 

9-54  127 
9-54  161 
9-54  195 
9-54  229 
9-54  263 
9-54  297 
9-54331 
9-54  365 
9-54  399 

o-43  074 
0-43  035 
0.42  996 

0.42  958 
0.42919 
0.42  880 
0.42  842 
0.42  803 
0.42  765 

9-97  201 

9-97  196 
9-97  192 

9-97  l87 
9.97  182 
9.97178 

9-97  173 
9.97  168 

9-97  l63 

9-54433 

33 
34 
34 
33 
34 
34 
33 

33 
34 
33 
34 
33 
34 
33 
33 
34 
33 
33 
33 
34 
33 
33 
33 
33 
33 
33 
33 
33 
33 

9-57  274 

0.42  726 

9-97  "59 

9-54  466 
9-54  5°° 
9-54  534 

9-54567 
9.54601 

9-54  635 
9-54  668 
9.54  702 
9-54  735 

9-57312 
9-57  351 
9-57  389 
9.57428 
9-57  466 
9-57  5°4 
9-57  543 
9-57  581 
9.57619 

0.42  688 
0.42  649 
0.42  611 
0.42572 
0.42  534 
0.42  496 

0-42  457 
0.42419 
0.42  381 

9-97  '54 
9-97  »49 
9  97  J45 
9,97  140 
9-97  !35 
9-97  130 
9.97  126 
9.97  121 
9.97116 

0 
2 

3 
4 
5 

0 
2 

3 

4 

5 

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40      39      38 

4.0     3.9     3.8 

12.0    II-7    II-4 
2O.O    19.5     ig.O 
28.0    27.3    26.6 
36.0    35.1    34.2 

544 

9-54  769 

9-57  658 

0.42  342 

9.97111 

9-97  I07 
9  97  102 

9-97  °97 
9.97092 
9.97087 
9-97  083 
9.97078 
9-97073 
9.97  068 
9.97  063 
9-97  °59 
9-97  °S4 
9-97  °49 
9-97  °44 
9-97  °39 
9-97  °35 
9.97030 

9-97  025 
9.97  020 

9.54  802 
9-54  836 
9-54  869 
9-54  903 
9-54936 
9-54  969 
9-55003 
9-55  036 
9-55  069 

9-57  696 
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9-57772 
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9-57  849 
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9-57  925 
9-57  963 
9.58001 

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0.42  304 
0.42  266 
0.42  228 
0.42  190 
0.42  151 
0.42113 

0.42  075 
0.42  037 
0.41  999 

9-55  I02 

0.41  961 

9-55  J36 
9-55  I69 
9-55  202 

9-55  235 
9-55  268 
9-55  3oi 
9-55  334 
9-55  367 
9-55  400 

9.58077 
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9-58  153 
9.58  191 
9.58  229 
9.58  267 
9.58  304 
9-58  342 
9.58  380 

0.41  923 
0.41  885 
0.41  847 
0.41  809 
0.41  771 
0.41  733 
0.41  696 
0.41  658 
0.41  620 

37      39     38 

3.7     4.9     4.8 
ii.  i   14.6  14.2 
18.5  24.4  23.8 
25-9  34-i  33-2 
33-3    —      — 

9-55433 

9.58418 

0.41  582 

9.97015 

L.  Cos. 

d. 

L.  Cot.    c.  d.    L.  Tan. 

L.  Sin.       d. 

' 

P.P. 

69' 


21° 


441 


L.  Sin.       d. 

L.  Tan.    c.  d.     L.  Cot. 

L.  Cos.    |   d. 

p.  p.          1 

0 

9-55  433 

9.58418 

0.41  582 

9.97015 

6O 

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9-55  466 

9-58455 

38 

0.41  543 

9.97010 

59 

2 

3 

9-55  499 
9-55  532 

33 

9-58493 
9-5853I 

38 

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0.41  507 
0.41  469 

9-97  005 
9.97  ooi 

4 

57 

38     37      36 

4 

9-55  564 

32 

9-58  569 

3° 

0.41  431 

9.96  996 

5 

56 

i 

3-8     3-7     3-6 

5 

9-55  597 

33 

9.58  606 

37 

0.41  394 

9.96991 

5 

55 

2 

7-6     7-4     7-2 

6 

9-55  630 

33 

9.58  644 

38 

0.41  356 

9.96  986 

S 

54 

3 

11.4  ii.  i   10.8 

7 
8 
9 
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9-55  663 
9-55  695 
9-55  728 

66 
32 
33 
33 

9.58681 
9-58  7'9 
9-58757 

61 
38 
38 
37 

0.41  319 
0.41  281 
0.41  243 

9.96981 
9.96976 
9.96971 

5 
5 
5 

53 
52 
51 
50 

4 

7 
8 

15.2  14.8  14.4 
19.0  18.5  18.0 

22.8    22.2    21.6 
26.6    25.9    25.2 
30.4    29.6    28.8 

9-55  76i 

9.58  794 

0.41  206 

9.96  966 

ii 

12 

9-55  793 
9.55  826 

33 

9.58  832 
9.58  869 

8 

0.41  1  68 
0.41  131 

9.96  962 
9-96957 

5 

49 
48 

9 

34-2  33-3  32-4 

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9-55  858 

32 

9.58  907 

3° 

0.41  093 

9.96952 

5 

47 

14 

9-55  891 

33 

9.58  944 

37 

0.41  056 

9.96  947 

5 

46 

33     32      31 

15 

9-55  923 

32 

9.58981 

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0.41  019 

9.96  942 

5 

45 

10 

9-55  956 

33 

9.59019 

3s 

0.40981 

9-96  937 

5 

44 

1 

3-3     j-2     3-1 

18 
19 

9-55  988 
9.56021 
9-56053 

33 
32 

9-59056 
9-59  094 
9-59I3I 

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38 

37 

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0.40  944 
0.40  906 
0.40  869 

9-96  932 
9-96927 
9.96  922 

5 

5 
5 

43 
42 
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3 
4 

5 

9-9     9-6    9-3 
13.2  12.8  12.4 
16.5   16.0  15  5 

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9.56  085 

9-59  168 

0.40  832 

9.96917 

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6 

19.8  19.2  18.6 

21 

9.56  118 

9-59  205 

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0.40  793 

9.96912 

39 

7 

23.1   22.4  21.7 

22 

9-56  150 

9-59  243 

0.40757 

9.96  907 

38 

s 

26.4  25.6  24.8 

23 

9.56  182 

32 

9-59  280 

37 

0.40  720 

9-96  903 

4 

37 

9 

29.7  28.8  27.9 

24 

9-56213 

33 

9-593I7 

37 

0.40  683 

9.96  898 

s 

36 

25 

9.56  247 

32 

9-59  354 

37 

0.40  646 

9.96  893 

5 

35 

26 

9.56279 

9-59  39  ! 

37 

0.40  609 

9.96  888 

5 

34 

654 

27 
28 
29 

3O 

9-56  343 
9-56  375 

32 
32 
32 
33 

9-59  429 
9-59  466 
9-59  5°3 

37 
37 
37 

0.40  571 
0.40  534 
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9.96  883 
9.96  878 
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9.96  868 

S 
5 
5 
5 

33 
32 
31 
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2 

3 

I  0.6    0.5     0.4 

1.2       1.0      0.8 

1.8     1.5     1.2 
2.4     2.0     i  6 

9.56408 

9-59  540 

0.40  460 

32 

9.56440 
9.56472 

32 

9-59577 
9.59614 

37 

0.40  423 
0.40  386 

9.96  863 
9.96858 

5 

5 

29 
28 

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3.0       2.5       2.0 
3.6       3.0       2.4 

33 

9.56  504 

9-59651 

0.40  349 

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5 

27 

7 

4-2     3-5     2.8 

34 

9-56  536 

32 

9-59  688 

37 

0.40312 

9.96  848 

5 

26 

8 

4.8    4.0    3.2 

35 

9.56568 

32 

9-59  723 

37 

0.40  275 

9-96  843 

5 

25 

9 

5-4    4-5     3-6 

36 

9-56  599 

9-59  762 

37 

0.40  238 

9-96  838 

5 

24 

37 

9.56631 

32 

9-59  799 

37 

0.40  201 

9-96  833 

5 

23 

38 

9-56663 

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0.40  165 

9.96  828 

5 

22 

39 

9-56  695 

32 

9-59872 

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0.40  1  28 

9-96  823 

21 

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9.56727 

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0.40  09  1 

9.96818 

20 

6         5         5 

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9-56759 

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9-96813 

19 

37       38      37 

42 
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9.56  790 
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32 

9-59  983 
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36 

0.40017 

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5 
5 

18 
17 

0 

3-i      3-8      3-7 

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9.60056 
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5 

5 

16 
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2 

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15.4    19.0    18.5 
21.6    26  6    25.9 

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9.56917 

9.60  130 

37 
16 

0.39  870 

9.96  788 

5 

4 

27.8    34.2    33.3 

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9.56949 

9.60  1  66 

o-39  834 

9.96  783 

13 

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33-9     —      — 

48 

9.56980 

31 

9.60  203 

37 

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9-96  778 

5 

12 

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9.57012 

9.60  240 

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0.39  760 

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50 

9-57044 

9.60  276 

0.39724 

9.96  707 

10 

5         4         4 

51 

9-57075 

9-60313 

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0.39  687 

9.96762 

9 

52 

9-57  I07 

32 

9.60  349 

3° 

0.39651 

9-96  757 

5 

8 

53 

9-57  138 

3 

9.60  386 

37 

0.39  614 

9-96  752 

5 

7 

3-6     4-8     4-6 

54 
55 
56 

9-57  169 
9-57  201 
9.57232 

31 

32 

9.60422 
9.60459 
9.60495 

3° 
37 
36 

o-39  578 
0.39  541 
o-39  5°5 

9.96  747 
9-96  742 
9-96  737 

5 
5 

5 

6 

5 
4 

2 

3 
4 

10.8    14.2    13.9 
18.0    23.8    23.1 
25.2    33.2    32.4 

57 

9-57  264 

32 

9.60  532 

37 
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0.39  468 

9.96  732 

S 

3 

5 

32-4     —      — 

58 

9-57  293 

9.60  568 

3° 

o-39  432 

9.96  727 

5 

2 

59 

9-57  326 

32 

9.60603 

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9.57358 

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9.96717 

0 

L.  Cos.      d. 

L.  Cot.    Ic.  d.     L,  Tan, 

L.  Sin.    i   d. 

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P.P. 

68° 


442 


22' 


L.  Sin. 

d. 

L.  Tan. 

c.  d 

L.  Cot. 

L.  Cos. 

d. 

P.  P. 

o 

9-57358 

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9.60  641 

36 

o-39  359 

9.96717 

6 

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2 

3 
4 

1 

9 

9-57  389 
9-57  420 
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9.57  482 
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9-57  6°7 
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31 
31 
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32 
31 
31 
31 
31 

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9.60677 
9.60714 
9.60  750 
9.60  786 
9.60  823 
9.60  859 
9.60895 
9.60931 
9.60  967 

11 

36 
37 
36 
36 
36 
36 

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0.39  286 
0.39  250 
0.39  214 
0.39177 
0.39  141 
0.39  105 
0.39  069 
0-39  033 

9.96711 
9.96  706 
9.96  701 
9.96  696 
9.96  691 
9.96  686 
9.96681 
9.96  676 
9.96  670 

5 
5 
5 
5 
5 
5 
5 
6 

59 
58 

57 
56 
55 
54 
53 
52 

2 

3 

4 
5 
6 

7 

37      36      35 

3-7     3-6     3-5 
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1  1.  1   10.8  10.5 
14.8  14.4  14.0 
18.5   18.0  17.5 

22.2    21.6    21.0 
25.9    25.2    24.5 

IO 

9-57  669 

9.61  004 

16 

0.38  99') 

9-96  665 

SO 

ii 

12 
13 

H 
15 
1  6 

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9-57  7°o 
9-57  73i 
9-57  762 

9-57  793 
9.57  824 

9-57  855 
9-57  885 
9.57916 
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31 
31 
31 
31 
31 
30 
31 
31 

9.61  040 
9.61  076 

9.61  112 
9.6l  148 
9.6l  184 
9.6l  22O 
9.6l  256 
9.6l  292 
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36 
36 
36 
36 
36 
36 
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0.38  960 
0.38  924 
0.38  888 
0.38852 
0.38816 
0.38  780 
0.38  744 
0.38  708 
0.38  672 

9.96  660 
9-96  655 
9.96  650 
9.96  645 
9.96  640 
9.96  634 
9.96  629 
9.96  624 
9.96619 

5 
5 
5 

i 

5 
5 
5 

49 
48 

47 
46 
45 
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4i 

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2 

3 

4 

5 

33-3  32-4  3I-5 

32      31      30 

3-2     3-1     3-0 
6.4     6.2     6.0 
9-6    9-3     9-0 

12.8    12.4    I2.O 

16.0  15.5   15.0 

2O 

9-57978 

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9-6  1  364 

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0.38  636 

9.96  614 

5 
6 

40 

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19.2  18.6  18.0 

21 
22 
23 
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3 

27 
28 
29 

9.58  008 
9-58  039 
9.58070 
9.58  101 

9-58131 
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9.58  192 
9-58  223 
9-58  253 

31 
31 
31 
30 
31 
3° 
31 
30 

31 

9.61  400 
9.61  436 
9.61  472 
9.61  508 
9.6  1  544 
9.61  579 
9.61  615 
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36 
36 
36 
36 

35 
36 
36 
36 

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0.38  600 
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0.38421 
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0.38  349 
0-38313 

9.96  608 
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9-96  593 
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9.96  582 

9.96577 
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9.96  567 

5 
5 

5 

5 
5 
5 

39 
38 
37 
36 
35 
34 
33 
32 
3' 

7 

S 

9 

2 

22.4  21.7  21.0 
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29      6       5 

2.9    0.6    0.5 

5-8      1.2      1.0 

87    i  8    ic 

3O 

9.58  284 

9:61  722 

l6 

0.38  278 

9.96  562 

30 

4 

1  1.  6    2.4    2.0 

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32 
33 

34 

3 

37 

9.583'4 
9-58  345 
9-58  375 
9.58406 
9-58436 
9.58467 

9-58  497 

31 
30 
31 
30 
31 
3° 

9.61  758 
9.61  794 
9.61  830 
9.61  865 
9.61  901 
9.61  936 
9.61  972 

36 
36 

35 
36 
35 
36 

0.38  242 
0.38  206 
0.38  170 

0-38  135 
0.38  099 
0.38  064 
0.38  028 

9-96556 
9-96551 
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9.96  541 
9-96  535 
9.96  530 

9.96  523 

5 
5 
5 
6 
5 
5 

29 
28 
27 
26 
25 
24 

21, 

5 
6 

7 
8 
9 

14.5    3-0    2.5 
17.4    3.6    3.0 
20.3    4-2    3-5 
23.2    4.8    4.0 
26.1    5.4    4.5 

38 
39 

9.58  527 
9-58  557 

3° 
30 

9.62  008 
9.62  043 

36 

36 

0.37  992 
0-37957 

9.96  520 
9.96514 

22 
21 

40 

9.58  588 

9.62  079 

0.37921 

9.96  509 

2O 

6        6 

4i 
42 
43 
44 

2 

47 
48 
49 

9.58618 
9.58  648 
9.58  678 
9.58  709 
9-58  739 
9.58  769 

9-58  799 
9.58829 
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30 
30 
3i 
30 
30 
30 
30 
3° 

9.62  114 
9.62  150 
9-62  185 

9.62  221 
9.62  256 
9.62  292 
9.62  327 
9.62  362 
9.62  398 

35 
36 
35 
36 

3 

35 

Ii 

0.37  886 
0.37  850 
o-37  815 
o-37  779 
o-37  744 
o-37  708 

o-37  673 
0.37  638 
0.37  602 

9.96  504 
9.96498 
9-96493 
9.96488 
9-96483 
9.96477 

9.96472 
9.96467 
9.96  461 

5 
6 

5 

5 

5 

I 

19 

18 
J7 
16 
15 
14 
i3 

12 
II 

36      35 

o 

3-o       2.9 
9-o      8.8 
,     15-0     14-6 

0      2I.O      20.4 
*      27.0      26.2 

6    33-o     32.1 

5O 

9.58889 

30 

9-62  433 

35 

037567. 

9.96456  j 

5 

10 

51 

52 
53 
54 
55 
56 

P 

59 

9.58919 
9-58  949 
9.58979 

9-59  009 
9-59  039 
9-59  069 
9-59  098 
9-59  128 
9-59  158 

30 
30 
30 
30 
3° 
29 

3° 
30 
30 

9.62  468 
9.62  504 
9.62  539 
9.62574 
9.62  609 
9.62  645 
9.62  680 
9.62715 
9.62  750 

36 
35 
35 

H 

35 
35 
35 

0-37  532 
0-37  496 
0.37461 

0.37  426 
0-37391 
o-37  355 
0.37  320 
o-37  285 
o-37  250 

9.96451   : 
9-96  445 
9.96  440 

9-96435 
9.96429 
9.96  424 
9.96419 
9-964I3 

9.96408  ; 

6 

5 
5 
6 

5 
5 
6 
5 

9 

8 

7 
6 

5 
4 
3 

2 
I 

o 
i 

2 

3 
4 

5 

555 

37      36      35 

3-7     3-6     3-5 
1  1.  1   10.8  10.5 
18.5   18.0  17.5 
25.9  25.2  24.5 
33-3  32-4  31-5 

60 

9-59  188 

9.62  785 

o-37  215 

9.96  403 

0 

L.  Cos. 

d. 

L.  Cot.  | 

c.  d. 

L.  Tan. 

L.  Sin. 

d. 

/ 

P.  P. 

67° 


23° 


44;! 


j  ' 

L.  Sin.   d. 

L.  Tan.  c.  d.j  L.  Cot. 

L.  Cos.  |  d. 

P.P. 

0 

2 

3 

4 

I 

7 
8 

9 
IO 

ii 

12 
13 

15 
17 

IS 

19 

20 

21 

22 
23 
24 
25 
26 

3 

29 
30 

32 
33 
34 
35 
36 

37 
38 
39 
4O 

41 

42 

43 
44 

49 
SO 

52 
53 
54 
55 
56 
57 
58 
59 
60 

9-59  1  88 

30 
29 
30 
30 
29 
30 
30 
29 
30 
29 

30 
29 
30 
29 

30 
29 
29 
30 
29 
29 

30 
29 
29 
29 
29 
30 
29 

29 
29 
29 
29 
29 
29 
29 
29 
29 
29 

29 
29 

28 

29 
29 
29 
28 
29 
29 
29 
28 

9.62  785 
9.62  820 
9.62  855 
9.62  890 
9.62  926 
9.62  961 
9.62  996 
9.63  031 
9.63  066 
9.63  101 

35 
35 
35 
36 
35 
35 
35 
35 
35 
34 
35 
35 
35 
35 
35 
35 
34 
35 
35 
35 
35 
34 
35 
35 
34 
35 
34 
35 
35 
34 
35 
34 
35 
34 
35 
34 
35 
34 
34 
35 
34 
34 
35 
34 
34 
35 
34 
34 
34 
34 
35 
34 
34 
34 
34 
34 
34 
34 
34 
34 

0-37  215 

9.96  403 

I  6 

5 
5 
6 

5 

i 

5 
6 

5 
5 
6 

I 

6 
1 

5 

5 
6 

5 
6 

5 
6 

5 
6 

5 
6 

I 

5 
6 
5 
6 

1 
I 

6 

I 

5 
6 

5 
6 

6 

5 
6 

5 
6 
6 

5 
6 

60 

59 
58 
57 
56 
55 
54 
53 
52 
51 
SO 
49 
48 
47 
46 
45 
44 

43 
42 

4O 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 

22 
21 

20 

19 
18 

*7 
16 
'5 
14 
13 

12 
I  I 

IO 

9 

8 

7 
6 
5 
4 
3 

2 

0 

36   35   34 

i   3-6  3-5  3-4 
2   7.2  7.0  6.8 
3  10.8  10.5  10.2 
4  14.4  14.0  13.6 
5  1  8.0  17.5  17.0 
6  21.6  21.0  20.4 
7  25.2  24.5  23.8 
8  28.8  28.0  27.2 
9  32-4  31-5  30.6 

30  29  28 

!    3.0   2.9   2.8 
2    6.0   5.8   5.6 

3   9.0  8.7  8.4 

4   I2.O  II.  (>  II.  2 

5  15.0  14.5  14.0 
6  18.0  17.4  16.8 
7  |  21.0  20.3  19.6 

S    24.0   23.2   22.4  , 

9  27.0  26.1  25.2 

6    5 

i  1  0.6   0.5 

2   1.2    1.0 

3  1-8   1.5 
4  2.4   2.0 
5  3-0   2.5 
6  3.6   3.0 
7  4-2   3-5 
8  4.8   4.0 
9  5-4   4-5 

9.59218 
9-59  247 
9-59  277 
9-59  307 
9-59  336 
9-59  366 
9-59  396 
9o9425 

9-59  484 

0.37  1  80 
°-37  H5 
0-37  n° 
0.37  074 

0-37  039 
0.37  004 

0.36  969 
0.36  934 
0.36  899 

9-96  397 
9.96  392 
9.96  387 
9.96  381 
9.96  376 
9.96  370 

9.96  365 
9.96  360 
9-96  354 

9-63  135 

0.36  865 

9-96  349 

9-59  5H 
9-59  543 
9-59573 
9.59  602 
9-59  632 
9-59  66  1 
9-59  690 
9-59  720 
9-59  749 

9-63  170 
9-63  205 
9-63  240 
9.63  275 
9.63  310 
9-63  345 

9-63  379 
9.63414 
9-63  449 
9.63  484 

0.36  830 
0-36  795 
0.36  760 

0.36  72? 
0.36  690 
0-36  655 
0.36621 
0.36  586 
0.36551 

9-96  343 
9-96  338 
9.96  333 
9.96327 
9.96  322 
9.96316 
9.96311 
9.96  305 
9.96  300 

9-59  778 

0.36516 

9.96  294 
9.96  289 
9.96  284 
9.96  278 

9-96  273 
9.96  267 
9.96  262 
9.96  256 
9.96251 
9.96  245 

9.59  808 
9-59  837 
9-59  866 

9-59  895 
9.59  924 
9-59  954 

9.59983 
9.60012 
9.60041 
9.60070 

9.63519 
9-63  553 
9-63  588 
9.63  623 
9-63657 
9.63  692 
9.63  726 
9.63  761 
9.63  796 

0.36481 
0.36  447 
0.36412 

0-36  377 
0.36  343 
0.36  308 
0.36  274 
0.36  239 
0.36  204 

9.63  830 

0.36  170 

9.96  240 

9.60  099 
9.60128 
9-6o  157 
9.60  1  86 
9.60  215 
9.60244 
9.60  273 
9.60  302 
9.60331 

9.63  865 
9.63  899 
9-63  934 
9-63  968 
9.64  003 
9.64  037 
9.64072 
9.64  106 
9.64  140 

0.36  135 
0.36  101 
0.36  066 
0.36  032 
0-35  997 
0-35  963 
0.35  928 

0.35  894 
0.35  860 

9.96  234 
9.96  229 
9-96  223 
9.96  218 
9.96212 
9.96  207 

9.96  201 
9.96  196 
9.96  190 

666 
36  35   34 

3.0  2.9  2.8 
\   9-0  8.8  8.5 
15.0  14.6  14.2 

J   21.0  2O-4  IU.X 
f   27.0  26.2  25.5 

§  33-o  32-1  31-2 

5    5 
35   34 

°i  3-5   34 

IO-5   10.2 
17.5   17.0 

•;  24.5  23-8 

5  3'-5  30-6 

9.60359 

9-64I75 

o-35  825 

9^6  185 

9.60  388 
9.60417 
9.60  446 
9.60  474 
9.60  503 
9-60532 
9.60  561 
9.60  589 
9.60618 

9.64  209 
9.64  243 
9.64  278 
9.64312 
9.64  346 
9.64  381 
9.64415 
9.64  449 
9-64  483 

o-35  79' 
0-35  757 
0-35  722 
0.35  688 
0-35  654 
0-35  6l9 
0-35  585 
0-35  55' 
0.35517 

9.96179 
9-96I74 

9.96  1  68 
9.96  162 
9-96  157 
9.96151 

9.96  146 
9.96  140 
9.96I3J 
9.96  129 

9.60  646 

29 

29 
28 
29 
28 
29 
28 
29 
28 
28 

9.64517 

o.35  483 

9.60  675 
9.60  704 
9.60  732 
9.60  761 
9.60  789 
9.60818 
9.60  846 
9-6o  875 
9-6o  903 

9.64  586 
9.64  620 
9.64  654 
9.64  688 
9.64  722 
9.64  756 
9.64  790 
9.64  824 

0.35  448 
0.354H 
0.35  38o 
0.35  346 
0.35312 
0.35  278 
0.35  244 
°-35  2I° 
Q-35  '76 
0.35  142 

9.96  123 
9.96  118 

9.96  112 
9.96  107 

9.96  ioi 
9.96  095 
9.96  090 
9.96  084 

9.96079 
9.96073 

9.60931 

9.64  858 

L.  Cos.   d.   L.  Cot.  jc.  d.  L.  Tan. 

L.  Sin.  !  d. 

f 

P.P. 

66° 


444 


24° 


, 

L.  Sin. 

d. 

L.  Tan. 

c.  d. 

L.  Cot. 

L.  Cos. 

d. 

p.p. 

0 

9.60931 

9-64  858 

0.35  142 

9.96073 

6 

60 

I 

2 

3 
4 

9 

9.60960 
9.60  988 
9.61  016 
9.61  045 
9.61  073 
9.61  101 
9.61  129 
9.61  158 
9.61  186 

28 
28 
29 
28 
28 
28 
29 
28 
28 

9.64  892 
9.64  926 
9.64  960 

9-64  994 
9.65028 
9.65  062 
9.65  096 
9.65  130 
9.65  164 

34 
34 
34 
34 
34 
34 
34 
34 

0.35  108 
0.35  074 
0.35  040 
0.35  006 
0.34972 
0-34  938 
0.34  904 
0.34  870 
0.34  836 

9.96  067 
9.96  062 
9.96  056 
9.96050 
9-96  045 
9.96  039 
9.96  034 
9.96028 
9.96  022 

5 
6 
6 

I 

5 
6 
6 

59 
58 
57 
56 
55 
54 
53 
52 
5i 

34    33 

i   34   3-3 
2   6.8   6.6 
3  10.2   9.9 
4  13.6  13.2 
5  17.0  16.5 

to 

9.61  214 

28 

9.65  197 

0.34  803 

9.96017 

6 

50 

6  20.4  19.8 

ii 

12 
13 
14 
15 

16 
17 

'9 

9.61  242 
9.61  270 
9.61  298 
9.61  326 
9-6i  354 
9.61  382 

9.61  411 

9.61  438 
9.61  466 

28 
28 
28 
28 
28 
29 

27 
28 
28 

9.65  231 
9.65  265 
9.65  299 

9-65  333 
9.65  366 
9.65  400 

9-65  434 
9.65  467 
9.65  501 

34 
34 
34 
33 
34 
34 
33 
34 

0.34  769 

0-34  735 
0.34  701 

0.34  667 

0-34  634 
0.34600 

0.34  566 
o-34  533 
0.34  499 

9.96011 
9.96  005 
9.96000 

9-95  994 
9.95  988 
9-95  982 
9-95  977 
9-95  97  ' 
9-95  965 

6 
5 
6 
6 
6 

5 
6 
6 

49 
48 

47 
46 
45 
44 
43 
42 
4i 

7  23.8  23.1 
8  27.2  26.4 
9  30.6  29.7 

29  28  27 

i   2.9  2.8  2.7 

20 

9.6  1  494 

28 

9-65  535 

0.34  465 

9.95  900 

6 

40 

2   5-8  5-6  5-4 

21 
22 
23 
24 
25 
26 

27 
28 
29 

9.61  522 
9-6i  550 
9.61  578 
9.61  606 
9.6  1  634 
9.61  662 
9.61  689 
9.61717 
9-6i  745 

28 
28 
28 
28 
28 

27 
28 
28 
28 

9.65  568 
9.65  602 
9.65  636 
9.65  669 
9.65  703 
9.65  736 
9.65  770 
9.65  803 
9-65  837 

34 
34 
33 
34 
33 
34 
33 
34 

0-34  432 
0-34  398 
0-34  364 

0-34331 

0.34  297 
0.34  264 
0.34  230 

0-34  197 
0.34  163 

9-95  954 
9-95  948 
9-95  942 
9-95  937 
9-95  93i 
9-95  925 
9-95  920 
9.95  914 
9-95  908 

6 
6 

5 
6 
6 

5 
6 
6 
6 

3 

37 
36 
35 
34 
33 
32 
3i 

3   8.7  8.4  8.1 
4  n.6  ii.  2  10.8 
5  14.5  14.0  13.5 
6  17.4  16.8  16.2 
7  20.3  19.6  18.9 
8  23.2  22.4  21.6 
9  26.1  25.2  24.3 

30 

9.61  773 

9.65  870 

0.34  130 

9-95  902 

30 

3i 
32 
33 
34 
35 
36 

3 

39 

9.61  800 
9.61  828 
9.61  856 
9.61  883 
9.61  911 
9.61  939 
9.61  966 
9.61  994 
9.62021 

28 
28 

27 
28 
28 
27 
28 
•27 
28 

9.65  904 

9-65  937 
9.65971 

9.66  004 
9.66  038 
9.66071 
9.66  104 
9.66138 
9.66  171 

33 

34 
33 
34 
33 
33 
34 
33 

0.34  096 
0.34  063 
0.34  029 

o-33  996 
0.33  962 
0.33  929 
0.33  896 
0.33  862 
0.33  829 

9-95  897 
9-95  89i 
9-95  88? 
9-95  879 
9-95  873 
9-95  868 
9-95  862 
9-95  856 
9-95  850 

6 
6 
6 
6 
5 
6 
6 
6 
6 

29 

28 

27 
26 
25 
24 
23 

22 
21 

6    5 

i  0.6   0.5 

2   1.2    1.0 

3  1.8   1.5 
4  2.4   2.0 
5  3-o   2.5 
6  3.6   3.0 
7  4-2   3-5 
8  4.8   4.0 

40 

9.62  049 

9.66  204 

0-33  796 

9-95  844 

2O 

9  5-4   4-5 

4i 

42 
43 
44 

9.62  076 
9.62  104 
9.62  131 
9.62  159 

28 
27 
28 

27 

9.66  238 
9.66271 
9.66  304 
9-66  337 

33 
33 
33 

0.33  762 

o-33  729 
0.33  696 

0-33  663 

9-95  839 
9-95  833 
9-95  827 
9.95821 

6 
6 
6 
6 

19 

18 
17 
16 

44i 

47 
48 

49 

9.62  186 
9.62  214 
9.62  241 
9.62  268 
9.62  296 

28 

27 
27 
28 

9.66371 
9.66  404 

9.66437 
9.66  470 
9-66  503 

33 
33 
33 
33 

0.33  629 
o-33  596 
o-33  563 
o-33  53° 
0-33  497 

9-95  815 
9.95  810 

9.95  804 
9-95  798 
9-95  792 

5 
6 
6 
6 

fi 

15 
H 
13 

12 
II 

6   6   j^ 

5O 

9.62  323 

9-66  537 

0-33  463 

9-95  786 

6 

IO 

5i 

52 
53 
54 
55 
56 

P 

59 

9.62  350 
9.62377 
9.62  405 
9.62  432 
9.62459 
9.62  486 
9.62513 
9.62  541 
9.62  568 

3 

27 
27 
27 
27 
28 
27 

9-66  570 
9.66  603 
9.66  636 
9.66  669 
9.66  702 
9-66  735 
9.66  768 
9.66801 
9.66  834 

33 
33 
33 
33 
33 
33 
33 
33 

0-33  430 
0-33  397 
0-33  364 
0.33331 
o-33  298 
0.33  265 

0.33  232 

0-33  199 
0.33  166 

9-95  780 
9-95  775 
9-95  769 
9-95  763 
9-95  757 
9-95  75  » 
9-95  745 
9-95  739 
9-95  733 

5 
6 
6 
6- 
6 
6 
6 
6 

9 
8 
7 
6 
5 
4 
3 

2 

I 

°    2.8   2.8   3-4 
8.5   8.2  10.2 

14.2  13.8  17.0 
•>  19.8  19.2  23.8 

J  25.5  24.8  30.6 

5  31.2  30.2  — 

60 

9-62  595 

9.66  867 

o-33  133 

9-95  728 

0 

L.  Cos. 

d. 

L.  Cot. 

c.  d. 

L.  Tan. 

L.  Sin. 

d. 

' 

p.p. 

65' 


25° 


445 


/ 

L.  Sin. 

d. 

L.  Tan. 

c.  d 

L.  Cot. 

L.  Cos. 

d. 

p.  p.          ! 

o 

9.62  505 

9.66  867 

0-33  133 

9-95  728 

6 

60 

I 

z 
3 
4 

1 

7 
8 

9 

9.62  622 
9.62  649 
9.62  676 
9.62  703 
9.62  730 
9.62  757 
9.62  784 
9.62811 
9.62  838 

27 
27 
27 
27 
27 
27 
27 
27 

9.66  900 

9-66933 
9.66  966 

9-66  999 
9.67  032 
9.67  065 
9.67  098 
9.67  131 
9.67  163 

33 
33 
33 

33 
33 
33 
33 
32 

0.33  loo 
0.33  067 
0-33  °34 
0.33001 
0.32  968 
0.32935 
0.32  902 
0.32  869 
0-32837 

9-95  722 
9-95  7i6 
9.95  710 

9-95  7°4 
9-95  698 
9.95  692 
9.95  686 
9.95  680 
9-95  674 

6 
6 
6 
6 
6 
6 
6 
6 
5 

59 
58 
57 
56 
55 
54 
53 
52 
Si 

33       32 

i       3-3       3-2 
2      6.6      6.4 
3      9-9      9-6 
4    13-2     12.8 
5     16.5     16.0 

10 

9.62  865 

9.67  196 

0.32  804 

9-95  668 

50 

6    19.8     19.2 

ii 

12 
«3 
H 

16 

!I 

19 

9.62892 
9.62918 
9.62  945 
9.62972 
9.62  999 
9.63  026 
9.63052 
9.63079 
9.63  106 

26 
27 
27 

2? 
27 
26 

27 
27 

9.67  229 
9.67  262 
9.67  295 
9.67  327 
9.67  360 
9-67  393 
9.67426 
9.67  458 
9.67491 

33 
33 
32 
33 

33 
33 
32 
33 

?? 

0.32771 
0-32  738 
0.32  705 

0.32  673 
0.32  640 
0.32  607 

0.32  574 
0.32542 
0.32  509 

9-95  663 
9-95  657 
9-95651 
9-95  64? 
9-95  639 
9-95  633 
9-95  627 
9-95  621 
9-95615 

5 
6 
6 
6 
6 
6 
6 
6 
6 
6 

49 
48 
47 
46 
45 
44 
43 
42 
4' 

7    23.1     22.4 
8    26.4    25.6 
9    29.7     28.8 

27        26 

i       2.7       2.6 
2      s  4      s  2 

20 

9-63  133 

26 

9.67  524 

32 

0.32476 

9-95  609 

6 

40 

3      8.1       7.8 

21 
22 
23 
24 
25 
26 

27 
28 
29 

9-63I59 
9.63  186 
9-63213 

9-63  239 
9.63  266 
9.63  292 

963319 
9-63  345 
9-63  372 

27 
27 
26 
27 
26 

27 
26 

z 

9-67  556 
9.67  589 
9.67  622 
9.67  654 
9.67  687 
9.67  719 

9-67752 
9.67  785 
9.67817 

33 
33 
32 
33 
32 
33 
33 
32 

3? 

0.32  444 
0.32411 
0.32  378 
0.32  346 

0.32313 
0.32  281 

0.32  248 
0.32215 
0.32  183 

9.95  603 
9-95  597 
9-95  59i 
9-95  585 
9-95  579 
9-95  573 
9-95  567 
9-95  56i 
9-95  55? 

6 
6 
6 
6 
6 
6 
6 
6 
6 

39 
38 
37 
36 
35 
34 
33 
32 
3i 

4    10.8     10.4 
5     '3-5     '3-0 
6     16.2     15.6 
7     18.9     18.2 
8    21.6     20.8 
9    24.3     23.4 

30 

9-63  398 

9.67  850 

0.32  150 

9-95  549 

6 

30 

765 

31 

32 
33 
34 

P 

37 
38 
39 

9-63  425 
9.63451 
9.63  478 
9.63  504 
9-63  53i 
9-63557 

9-63  583 
9.63  610 
9.63  636 

26 
27 
26 

27 
26 
26 

z 

->6 

9.67  882 
9.67915 
9.67  947 
9.67  980 
9.68012 
9.68  044 
9.68077 
9.68  109 
9.68  142 

33 
32 
33 
32 
32 
33 
32 
33 

32 

0.32  118 
0.32  085 
0.32053 
0.32  020 
0.31  988 
0.31  956 
0.31  923 
0.31  891 
0.31  858 

9-95  543 
9-95  537 
9-95531 
9-95  525 
9-95  519 
9-95513 
9-95  5°7 
9-95  5°° 
9-95  494 

6 
6 
6 
6 
6 
6 

7 
6 
6 

29 
28 
27 
26 
25 
24 
23 

22 
21 

i  |  0.7     0.6    0.5 

2  1   1.4       1.2       I.O 

3    2.1     1.8     1.5 
4    2.8    2.4    2.0 

5    3-5     3-o    2.5 
6    4.2    3.6    3.0 
7    4-9    4-2    3-5 
8    5.6    4.8    4.0 
9    6.3     5.4    4.5 

4O 

9.63  662 

27 

9.68174 

32 

0.31  826 

9-95  488 

6 

20 

4i 

42 
4-2 

9.63  689 

9-637I5 
n  6^  741 

26 
26 

9.68  206 
9-68  239 
9.68  271 

33 

32 

0.31  794 
0.31  761 
0.31  729 

9-95  482 
9-95  476 
9-95  47° 

6 
6 

'9 
18 
17 

44 
45 
46 

47 
48 
49 

9.63  767 

9-63  794 
9.63  820 

9.63  846 
9.63  872 
9.63  898 

26 

27 
26 
26 
26 
26 

"6 

9.68  303 
9.68  336 
9.68  368 
9.68  400 
9.68432 
9-68465 

32 
33 
32 
32 
32 
33 

32 

0.31  697 
0.31  664 
0.31  632 
0.31  600 
,  0.31  568 
0-3'  535 

9-95  464 
9-95  458 
9-95  452 
9-95  446 
9-95  440 
9-95  434 

I 

6 
6 
6 
6 

16 
'5 
14 
13 

12 
II 

766 
32      32       33 

50 

9.63  924 

"6 

9-68  497 

72 

0.31  5°3 

9-95  427 

6 

10 

,     2.3      2.7      3.3 

5' 
52 
53 
54 

II 

57 
58 
59 

9-63  95° 
9.63976 
9.64  002 
9.64028 
9.64054 
9.64  080 
9.64  1  06 
9.64  132 
9.64  158 

26 
26 
26 
26 
26 
26 
26 
26 
•>6 

9-68  529 
9.68  561 
9.68  593 
9.68  626 
9.68658 
9.68  690 
9.68  722 
9-68  754 
9.68  786 

32 
32 

33 
32 
32 
32 
32 
32 

1.2 

0-31  47  i 
0.31  439 
0.31  407 

0-3'  374 
0.31  342 
0.31  310 

0.31  278 
0.31  246 
0.31  214 

9.95  421 
9-954I5 
9-95  409 
9-95  403 
9-95  397 
9-95391 
9-95  384 
9-95  378 
9-95  372 

6 
6 
6 
6 
6 

7 
6 
6 

6 

9 
8 

7 
6 
5 
4 
3 

2 

11.4    13.3    16.5 
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4  20.6    24.0    29.7 

j  '£  *9-3  - 

60 

9.64  184 

9.68818 

0.31  182 

9-95  366 

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L.  Cos. 

d. 

L.  Cot. 

c.  d. 

L.  Tan. 

L.  Sin. 

d. 

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P.P. 

64° 


446 


26° 


1  , 

L.  Sin. 

d. 

L.  Tan. 

c.  d. 

L.  Cot. 

L.  Cos. 

d. 

p.  p. 

o 

9.64  184 

26 

9.68818 

32 

0.31  182 

9-95  366 

6 

6O 

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•2 

3 

4 

1 
1 

9 

9.64  2IO 
9.64  236 
9.64  262 
9.64  288 
9-643I3 

9-64  339 
9.64  365 
9.64  391 
9.64417 

26 
26 
26 

25 
26 
26 
26 
26 

9.68  850 
9.68  882 
9.68914 
9.68  946 
9.68  978 
9.69010 
9.69  042 
9.69  074 
9.69  106 

32 
32 
32 
32 
32 
32 
32 
32 

0.31  150 
0.31  1  18 
0.31  086 
0.31  054 

0.31  022 
0.30  990 
0.30958 
0.30  926 
0.30  894 

9-95  36o 
9-95  354 
9-95  348 
9-95  34  i 
9-95  335 
9-95  329 
9-95  323 
9-953I7 
9-95  3io 

6 
6 

7 
6 
6 
6 
6 

I 

59 
58 
57 
56 
55 
54 
53 
52 
5i 

32    31 

i   3-2   3-1 

2    6.4    6.2 

3   9-6   9-3 
4  i  12.8   12.4 
5  16.0   15.5 

!  10 

9.64  442 

26 

9.69  138 

0.30  862 

9-95  3°4 

6 

50 

ii 

12 
13 

H 
15 

16 

\l 

19 

9.64  468 
9.64  494 
9.64  519 

9-64  545 
9-64571 
9.64  596 

9.64  622 
9.64  647 
9.64  673 

26 
25 
26 
26 
25 
26 

25 
26 

9.69170 
9.69  202 
9.69  234 
9.69  266 
9.69  298 
9.69  329 
9.69  361 
9-69  393 
9.69  425 

32 
32 
32 
32 
3i 
32 
32 
32 

0.30  830 
0.30  798 
0.30  766 

0.30  734 
0.30  702 
0.30  671 
0.30  639 
0.30  607 
°-3°  575 

9.95  298 

9-95  292 
9.95  286 

9-95  279 
9-95  273 
9.95  267 

9-95  261 
9-95  254 
9-95  248 

6 
6 

7 
6 
6 
6 

7 
6 
6 

49 
48 
47 
46 
45 
44 
43 
42 
41 

8  25.6  24.8 
9  |  28.8   27.9 

26  25   24 

i  |  2.6  2.5  2.4 

2    5.2   5.0   4.8 

20 

9.64  698 

26 

9-69  457 

0-30  543 

9-95  242 

5 

40 

3   7.8  7.5  7.2 

21 
22 
23 
24 

3 
3 

29 

9.64  724 
9.64  749 
9-64  775 
9.64  800 
9.64  826 
9.64851 
9.64  877 
9.64  902 
9.64  927 

25 
26 

25 
26 
25 
26 

25 
25 

9.69  488 
9.69  520 
9-69  SS2 
9.69  584 
9.69615 
9.69  647 
9.69  679 
9.69  710 
9.69  742 

32 
32 
32 
3i 
32 
32 
3i 
32 

0.30512 
0.30  480 
0.30  448 
0.30416 
0.30  385 
0.30  353 
0.30321 
0.30  290 
0.30  258 

9-95  236 
9.95  229 
9-95  223 
9.95217 
9.95211 
9-95  204 
9-95  "98 
9-95  192 
9-95  l85 

\ 

6 

6 
7 
6 
6 

7 
5 

39 
38 
37 
36 
35 
34 
33 
32 
3i 

4  10.4  10.0  9.6 
5  13.0  12.5  12.0 
6  15.6  15.0  14.4 
7  18.2  17.5  16.8 
8  20.8  20.0  19.2 
9  23.4  22.5  21.6 

30 

9-64  953 

9.69  774 

3- 
•ji 

0.30  226 

9-95  179 

6 

3O 

7    6 

3i 
32 
33 
34 

P 

37 
38 
39 

9.64  978 
9.65  003 
9.65  029 

9-65  °54 
9.65  079 
9.65  104 

9-65  13° 
9-65  155 

9.65  i  So 

11 

25 
25 
25 
26 

25 

25 

9.69  805 
9.69  837 
9.69  868 
9.69  900 
9.69  932 
9.69  963 

9-69  995 
9.70026 
9.70058 

32 
3i 
32 

32 
3i 
32 
31 

32 

•M 

0.30  193 
0.30  163 
0.30  132 
0.30  100 
0.30  068 
0.30037 
0.30  005 
0.29  974 
0.29  942 

9-95  *73 
9-95  l67 
9-95  I6° 
9-95  J54 
9-95  H8 
9-95  H1 
9-95  "35 
9-95  I29 

9-95  "2 

6 
7 
6 
6 

I 

7 
6 

29 
28 

27 
26 
25 
24 
23 

22 
21 

i  0.7   0.6 

2   1.4    1.2 

3  2.1   1.8 
4  2.8   2.4 
5  3-5   3-o 
6  4.2   3.6 
7  4-9   4-2 
8  5.6   4.8 
9  6.3   5.4 

4O 

9.65  205 

9.70  089 

0.29911 

9-95  II6 

6 

20 

4i 
42 

9-65  230 
9-65  255 

11 

9.70  121 
9.70152 

31 
32 

0.29  879 
0.29  848 

9-95  II0 
9-95  I03 

7 
6 

19 
18 

43 
44 

4I 
46 

47 
48 

49 

9.65  306 
9-65  33' 
9-65  356 
9.65  381 
9.65  406 
9-6543I 

25 

25 
25 
25 
25 
25 

9.70215 
9.70  247 
9.70  278 
9.70  309 
9.70341 
9.70372 

31 
32 
31 
31 
32 
31 

0.29  785 
0.29  753 
0.29  722 
0.29  691 
0.29  659 
0.29  628 

9-95  °9Q 
9-95  °84 
9.95  078 

9.95071 
9-95  o6? 
9-95  °59 

7 
6 
6 

7 
6 
6 

17 
16 
J5 
H 
13 

12 
II 

776 
32  31   32 

SO 

9-65  456 

9.70  404 

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0.29  596 

9-95  °52 

6 

IO 

°    2.3   2.2   2.7 

51 

52 
53 
54 
55 
56 

H 

59 

9.65  481 
9.65  506 
9-65  531 

9.65  556 
9.65  580 
9.65  605 
9.65  630 
9-65  655 
9.65  680 

25 
25 
25 
24 
25 
25 
25 
25 

9.70435 
9.70  466 
9.70  498 
9.70  529 
9.70  560 
9.70  592 
9.70  623 
9.70654 
9.70  685 

3i 
32 
3i 
31 
32 
31 
3i 
3i 

0.29  565 
0.29  534 
0.29  502 
0.29471 
0.29  440 
0.29408 
0.29  377 
0.29  346 
0.29315 

9-95  046 
9-95  °39 
9-95  °33 

9-95  027 
9.95  020 

9-95  OI4 
9-95  °°7 
9-95  ooi 
9-94  993 

I 

6 

7 
6 

7 
6 
6 

I 

7 
6 
5 
4 
3 

2 

1   6.9  6.6  8.0 
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»  1  6.0  15.5  18.7 
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5  25.1  24.4  29.3 
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9-65  7°5 

9.70717 

0.29  283 

9-94  988 

0 

L.  Cos. 

d. 

L.  Cot. 

c.  d. 

L.  Tan. 

L.  Sin. 

d. 

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P.P. 

63e 


27" 


447 


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L.  Sin. 

d. 

L.  Tan. 

c.  d. 

L.  Cot. 

L.  Cos. 

d. 

p.p. 

0 

9.65  705 

24 

9.70717 

0.29  283 

9-94  988 

6 

60 

2 

3 

4 

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7 
8 

9 

9.65  729 
9-65  754 
9-65  779 
9.65  804 
9.65  828 
9-65  853 
9.65  878 
9.65  902 
9.65  927 

25 

25 
25 
24 
25 
25 
24 
25 

9-7°  748 
9.70779 
9.70810 
9.70841 
9.70873 
9.70  904 

9-70933 
9.70966 
9.70997 

3i 
31 
31 
32 

0.29  252 

0.29*21 
0.29  190 
0.29  I59 
0.29  127 
O.29  096 
O.29  065 
0.29034 
0.29  003 

9.94982 
9-94  975 
9-94  969 
9-94  962 
9-94  956 
9-94  949 
9-94  943 
9-94  936 
9-94  93° 

7 
6 

7 
6 
7 
6 

7 
6 

59 
58 

57 
56 
55 
54 
53 
52 
51 

32      31      30 

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2         6.4      6.2     '6.0 

3      9-6    9-3     9-o 
4     12.8  12.4  12.0 
5     16.0  15.5  15.0 

IO 

9-65  952 

9.71  028 

0.28  972 

9.94923 

6 

50 

ii 

12 

'3 
14 

11 

17 
18 
19 

9.65  976 
9.66  ooi 
9.66025 
9.66  050 
9.66  075 
9.66  099 
9.66124 
9.66  148 
9.66  173 

25 

24 

25 
25 
24 
25 
24 
25 

9-71  °59 
9.71090 
9.71  121 

9-71  X53 
9.71  184 
9.71  215 
9.71  246 
9.71  277 
9.71  308 

31 

32 

31 

0.28  94! 
0.28910 
0.28  879 
0.28  847 
0.28816 
0.28  785 

0.28  754 
0.28  723 
0.28  692 

9.94917 
9.94911 
9-94  904 
9-94  898 
9.94891 
9.94  885 
9.94878 
9.94871 
9.94865 

6 
7 
6 

I 

7 
7 
6 

49 
48 

47 
46 
45 
44 
43 
42 

8    25.6  24.8  24.0 
9    28.8  27.9  27.0 

25      24      23 

i       2.5     2.4     2.3 

2         5.0      4.8      4.6 

2O 

9.66  197 

24 

9-7i  339 

7T 

0.28  661 

9-94858 

6 

4O 

3      7-5     7-2    6.9 

21 

22 
23 
24 

3 

27 
28 

29 

9.66  221 
9.66  246 
9.66  270 
9-66  295 
9.66319 

9-66  343 
9.66  368 
9.66  392 
9.66416 

25 
24 
25 
24 
24 
25 
24 
24 

9.71  370 
9.71  401 
9-71  43i 
9.71  462 

9-71  493 
9.71  524 

9-7i  555 
9.71  586 
9.71  617 

30 
31 

31 
31 
31 

0.28  630 
0.28  599 
0.28  569 
0.28  538 
0.28  507 
0.28476 
0.28  445 
0.28414 
0.28  383 

9-94852 
9-94  845 
9-94  839 
9-94  832 
9.94826 
9.94819 
9.94813 
9.94  806 
9-94  799 

7 
6 

7 
6 
7 
6 

7 

7 
5 

P 

37 
36 
35 
34 
33 
32 

5     12.5  12.0  11.5 
6    15.0  14.4  13.8 
7     17.5   16.8  16.1 
8    20.0  19.2  18.4 
9    22.5  21.6  20.7 

3O 

9.66441 

9.71  648 

0.28  352 

9-94  793 

3O 

7       6 

32 
33 
34 

P 

37 
38 
39 

9.66  465 
9.66489 
9.66513 
9.66  537 
9.66  562 
9.66  586 
9.66610 
9.66  634 
9.66658 

24 
24 
24 
25 
24 
24 
24 
24 

9.71  679 
9.71  709 
9.71  740 
9.71  771 
9.71  802 
9-71  833 
9.71  863 
9.71  894 
9.71925 

30 
30 

0.28  321 
0.28  291 
0.28  260 
0.28  229 
0.28  198 
0.28  167 
0.28137 
0.28  106 
0.28075 

9-94  786 
9.94  780 

9-94  773 
9-94  767 
9-94  760 
9-94  753 
9-94  747 
9-94  740 
9-94  734 

6 
7 
6 

7 
7 
6 

I 

29 
28 
27 
26 
25 
24 
23 

22 
21 

i     0.7     0.6 

2       1.4       1.2 

3    2.1     1.8 

4    2.8     2.4 

5    3-5     3-° 
6    4.2     3.6 
7    4.9    4.2 
8    5.6    4.8 
9    6.3     5.4 

4O 

9.66  682 

9-7r955 

0.28  045 

9.94  727 

20 

42 

9.66  706 
9.66731 

25 

24 

9.71986 
9.72017 

| 

0.2,8014 
0.27  983 

9-94  720 
9.94714 

6 

19 
18 

43 
44 
45 
46 

11 

49 

9-°°  755 
9.66779 
9.66  803 
9.66827 
9.66851 
9.66875 
9.66  899 

24 
24 

24 
24 
24 
24 

9.72078 
9.72  109 
9.72  140 
9.72  170 

9.72  201 

30 

30 
30 

0.27  922 
0.27  891 
0.27  860 
0.27  830 
0.27  799 
0.27  769 

9-94  7°° 
9-94  694 
9-94  687 
9.94  680 
9-94  674 
9-94  667 

7 
6 

7 

6 

7 

16 

'5 
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13 

12 
II 

766 
30     31      30 

o 

50 

9.66  922 

9.72  262 

0.27  738 

9.94  660 

IO 

i      *•'     f'|    f'5 

51 

52 
53 
54 
55 
56 

P 

59 

9.66  946 
9.66970 
9.66  994 
9.67018 
9.67  042 
9.67066 
9.67  090 
9.67113 
9.67  137 

24 
24 
24 
24 
24 
24 
23 
24 

9.72  293 

9-72354 
9.72384 
9.72415 

9-72  445 
9.72476 
9.72  506 
9-72  537 

30 
3° 

3° 
3' 

30 

30 

0.27  707 
0.27  677 
0.27  646 
0.27  616 
0.27  585 
0-27  553 
0.27  524 
0.27  494 
0.27  463 

9-94  654 
9.94  647 
9.94  640 

9-94  634 
9-94  627 
9.94  620 
9.94614 
9-94  607 
9.94  600 

7 
7 
6 

7 
7 
6 

7 
7 
7 

9 
8 

7 
6 
5 

4 
3 

2 
I 

6.4    7.8    7.5 
10.7  12.9  12.5 
•*     15.0  18.1   17.5 
*     19.3  23.2  22.5 
5    23.6  28.4  27.5 
o    27.9    —     — 

60 

9.67  161 

9.72567 

0.27  433 

9-94  593 

o 

L.  Cos. 

d. 

L.  Cot. 

c.  d 

|    L.Tan. 

L,  Sin. 

d. 

' 

P.P. 

62° 


448 


28C 


L.  Sin. 

d. 

L.  Tan. 

c.  d. 

L.  Cot. 

L.  Cos. 

d. 

p.p. 

o 

9.67  161 

9-72567  !  „ 

0-27433 

9-94  593      fi 

6O 

I 

9-67  185 

9-72598 

0.27  402 

9-94  587 

59 

2 

9.67  208 

23 

9.72  628 

3° 

0-2737* 

9-94  580 

S8 

3 

9.67  232 

9.72659 

31 

0.27  341 

9-94  573 

57 

4 

9.67  256 

24 

9.72  689 

3° 

0.27311 

9-94567' 

31      30     29 

9.67  280 
9-67  303 

24 
23 

9.72  720 

9-72  75° 

31 
30 

0.27  280 
0.27  250 

9-94  560 
9-94553 

7 

7 

55 
54 

i       3.1     3.0     2.9 

2         6.2       6.0       5.8 

7 

9-67  327 

9.72  780 

0.27  220 

9.94  546 

53 

3      9.3    9.0     8.7 

8 

9-67  35° 

23 

9.72  811 

0.27  189 

9-94  540 

S2 

4    12.4  12.0  1  1.  6 

9 

9-67  374 

9.72841 

0.27  159 

9-94  533 

5* 

5     J5-5  J5-o  H-5 

IO 

9.67  398 

9.72872 

0.27  128 

9-94526 

5O 

6    18.6  18.0  17.4 

ii 

12 
13 

9.67421 
9.67  445 
9.67  468 

24 
23 

9.72  902 
9.72932 
9-72  963 

30 
31 

0.27  098 
0.27068 

9-94  5  '9 
9-94  5  J3 
9.94  506 

6 

7 

49 
48 

47 

7    21.7  21.0  20.3 
8    24.8  24.0  23.2 
9    27.9  27.0  26.1 

14 

9.67  492 

24 

9.72993 

3° 

0.27OO7 

9.94  499 

7 

46 

15 

9.67515 

23 

9-73023 

3° 

0.26977 

9-94  492 

7 

45 

16 

9-67  539 

973054 

31 

O.26  946 

9-94  485 

44 

18 

9.67  562 
9.67  586 

24 

9.73084 
9-73  "4 

3° 
30 

0.26916 

0.26  886 

9-94  479 
9.94472 

7 

43 
42 

24      23      22 

19 

9.67  609 

9-73  H4 

3° 

0.26856 

9-94  465 

7 

41 

I          2.4       2.3       2.2 

2O 

21 
22 
23 

24 

3 

9-67  633 

23 
24 
23 
23 
24 
23 

9-73  175 
9-73  205 
9-73  235 
9-73  265 
9-73  295 

9-73356 

30 
30 
30 
30 

30 

0.26  825 

9-94  458 

7 
6 
7 
7 
7 
7 

4O 

39 
38 
37 
36 
35 
34 

2         4.8      4.6      4.4 

3      7.2     6.9     6.6 
4      9.6    9.2     8.8 
5    12.0  11.5  n.o 
6     14.4  13.8  13.2 
7     16.8  16.1   15.4 
8     19.2  18.4  17.6 
9    21.6  20.7  19.8 

9.67  656 
9.67  680 
9.67  703 
9.67  726 
9.67  730 
9-67  773 

0.26  795 
0.26  765 
0.26  735 
0.26  703 
0.26674 
0.26644 

9-94  45  * 
9.94  443 
9-94438 
9-94431 
9-94  424 
9.94417 

27 

9.67  796 

23 

9-73  386 

3° 

0.26614 

9.94410 

S3 

28 

9.67  820 

24 

9.73416 

3° 

0.26  584 

9-94  404 

32 

29 

9-67  843 

23 

9-73446 

3° 

0.26  554 

9-94  397 

31 

3O 

9.67  866 

973476 

0.26524 

9-94  390 

SO 

31 

9.67  890 

9-73  5°7 

0.26493 

9-94  383 

29 

7          6 

32 

9.67913 

23 

9-73537 

3° 

0.26  463 

9-94  376 

28 

, 

33 

9.67  936 

23 

9-73  567 

3° 

0.26433 

9-94  369 

7 

27 

2      1-4         1.2 

34 

9.67  959 

23 

9-73597 

0.26403 

9.94  362 

26 

3    2.1       1.8 

35 

9.67  982 

23 

9.73627 

3° 

0.26373 

9-94  355 

g 

25 

4    2.8       2.4 

36 

9.68  006 

9-73  657 

0.26343 

9-94  349 

24 

5    3-5       o-o 

37 

9.68  029 

23 

9-73  687 

3° 

0.26313 

9-94  342 

7 

23 

6    4.2       3.6 

9.68  052 

23 

9-737I7 

3° 

0.26  283 

9-94  335 

7 

22 

7    4-9      4-2 

39 

9.68075 

23 

9-73  747 

3° 

0.26  253 

9-94  328 

7 

21 

8    5.6      4.8 

40 

9.68  098 

9-73  777 

0.26  223 

9  94  321 

2O 

9    6.3       5.4 

41 

9.68  121 

9-73  807 

0.26  193 

9-943I4 

19 

42 

9.68  144 

23 

9-73  837 

3° 

0.26  163 

9-94  307 

7 

18 

43 

9.68  167 

23 

9.73867 

3° 

0.26  133 

9-94  300 

7 

17 

44 

9.68  190 

*3 

9-73  897 

3° 

0.26  103 

9-94  293 

i 

16 

45 

9.68  213 

23 

9-73927 

3° 

0.26073 

9.94  286 

15 

46 

9-68  237 

9-73957 

3° 

0.26043 

9-94  279 

47 

9.68  260 

23 

973987 

3° 

0.26013 

9-94  273 

J3 

48 

9.68  283 

23 

9.74017 

3° 

0.25  983 

9.94  266 

7 

12 

766 

49 

9-68  305 

9.74047 

3° 

0.25  953 

9-94  259 

7 

ii 

31      31      30 

50 

9.68  328 

9.74077 

0.25  923 

9.94  252 

10 

0                           , 

51 

52 
53 
54 
55 
56 

9.68351 
9-68  374 
9-68  397 
9.68  420 
9  .68  443 
9.68  466 

23 
23 
23 
23 
23 

9-74  107 
9-74  137 
9.74  1  66 

9-74  196 
9.74  226 
9.74256 

30 

29 

30 
30 
30 

0.25  893 
0.25  863 
0.25  834 
0.25  804 
0-25  774 
0.25  744 

9-94  245 
9-94  238 
9.94  231 

9-94  224 
9.94217 
9.94210 

7 
7 

7 
7 
7 

9 
8 

7 
6 
5 
4 

2.2      2.0      2.5 

1      6.6     7.8     7.5 
ii.  i   12.9  12.5 
•     15.5   18.1   17.5 
J     19.9  23.2  22.5 
5    24.4  28.4  27.5 
6     28.8    _      _    I 

S7 

9.68  489 

23 

9.74  286 

3° 

0.25714 

9  94  203 

7 

3 

7 

58 

9.68512 

9-74  3l6 

0.25  684 

9.94  196 

7 

2 

59 

9-68  534 

23 

9-74  345 

3° 

0-25  653 

9-94  '89 

7 

I 

60 

9.68557 

"•7-1  575 

0.25  623 

9.94  182 

0 

L.  Cos.       d. 

L.  Cot.   |c.  d.    L.  Tan. 

L.  Sin.    |  d. 

' 

P.P. 

61° 


29° 


449 


' 

L.  Sin. 

d. 

L.  Tan. 

c.  d. 

L.  Cot. 

L.  Cos. 

d. 

p.  p. 

!  0 

9-68557 

2? 

9-74  375 

3O 

0.25  625 

9.94  182 

60 

I 

2 

3 

4 

I 

I 

9 

9.68  580 
9.68  603 
9.68  625 
9.68  648 
9.68671 
9.68  694 
9.68  716 
9.68  739 
9.68  762 

23 

22 

23 
23 
23 

22 

23 

23 

9-74  405 
9-74  435 
9-74465 
9-74  494 
9-74  524 
9-74  554 
9-74583 
9-74613 
9-74  643 

30 
3° 
29 

0.25  595 
0-25  565 
0-25  535 
0.25  506 
0.25  476 
0.25  446 
0.25417 
0.25  387 

9.94I75 
9.94  1  68 
9.94  161 

9-94  154 
9-94  H7 
9.94  140 

9-94  133 
9.94126 
9.94119 

7 

7 
7 
7 
7 
7 
7 
7 

59 

57 
56 
55 
54 
53 
52 

30  29  23 

i   3.0  2.9  2.3 

2    6.0   5.8   4.6 

3   9.0  8.7  6.9 
4  120  1  1.  6  9.2 
5  15.0  14.5  11.5 
0  iS.o  17.4  13.8 

IO 

9.68  784 

9-74  673 

0.25  327 

9.94112 

50 

7  21.0  20.3  16.1 

1  1 

12 
13 
H 

16 
19 

9.68  807 
9.68  829 
9.68852 
9.68  875 
9.68  897 
9.68  920 
9.68  942 
9.68  965 
9.68  987 

22 
23 
23 
22 
23 
22 

23 
22 
2T. 

9.74  702 
9-74  732 
9.74  762 

9.74  791 
9.74821 
9-7485I 
9.74880 
9.74910 
9-74  939 

30 
30 

30 
29 
30 
29 

0.25  298 
0.25  268 
0.25  238 
0.25  209 
0.25  179 
0.25  149 

O.25  120 
O.25  090 

0.25  061 

9.94  105 
9.94  098 
9.94  090 

9-94  083 
9.94076 
9-94  069 
9.94  062 
9-94  05? 
9-94  048 

7 
8 

7 
7 

7 
7 

7 
7 

49 
48 
47 
46 
45 
44 
43 
42 

8  24.0  23.2  18.4 
9  27.0  26.1  20.7 

22  8   7 

I    2.2  0.8  0.7 

2   4.4  1.6  1.4 

120 

9.09010 

22 

9.74909 

0.25031 

9.94041 

40 

4   8.8  3.2  2.8 

21 
22 
23 
24 

25 
26 

27 
28 
29 

9.69  032 
9.69  053 
9.69  077 
9.69  loo 

9.69  122 
9.69  144 
9.69  167 
9.69  189 
9.69  212 

23 
22 

23 
22 
22 

23 
22 
23 

9-74  998 
9.75028 
9-75058 

9-75  087 
9-75  "7 
9-75  146 
9.75  176 
9-75  205 
9-75  235 

30 
30 
29 
30 
29 

29 
30 

0.25  002 
0.24  972 
0.24  942 
0.24913 
0.24  883 
0.24  854 
0.24  824 

0.24  795 
0.24  765 

9-94034 
9-94  027 
9.94  020 
9.94012 
9.94  005 
9-93  998 
9-93991 
9-93  984 
9-93  977 

7 
7 
8 

7 
7 

7 

39 
38 
37 

34 
33 
32 

5  1  1.0  4.0  3.5 
6  13.2  4.8  4.2 
7  15.4  5.6  4.9 
8  17.6  6.4  5.6 
9  19.8  7.2  6.3 

3O 

9.69  234 

22 

9-75  264 

29 

0.24  736 

9-93  970 

30 

32 
34 

P 

37 
39 

9.69  256 
9.69  279 
9.69  301 

9-69  323 

9-69  345 
9.69  368 

9.69  390 
9.69412 
9-69  434 

23 
22 
22 
22 
23 
22 
22 
22 
22 

9-75  294 
9-75  323 
9-75  353 
9-75  382 
9-754" 
9-75441 

9-75  470 
9.75  500 

9-75  529 

29 
30 
29 
29 
30 
29 
30 
29 

0.24  706 
0.24677 
0.24  647 
0.24618 
0.24  589 
0.24  559 
0.24  530 
0.24  500 
0.24471 

9-93  963 
9-93  955 
9.93  948 

9-93  94  ! 
9-93  934 
9-93  927 
9-93  920 
9.93912 
9-93  905 

7 

7 

7 
7 
8 

7 

29 
28 
27 
26 
25 
24 
23 

22 
21 

8    8 
30   29 

0   1.9   1.8 
\   5-6   5-4 
9-4   9-i 
:  13.1  12.7 

40 

9.69  456 

9-75  558 

0.24  442 

9-93  898 

20 

:  16.9  16.3 

4i 

42 

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44 
45 
46 

47 
49 

9.69  479 
9.69  501 
9.69523 

9-69  545 
9.69  567 
9.69  589 
9.69  611 
9-69  633 
9-69  655 

22 
22 
22 
22 
22 
22 
22 
22 

9-75  588 
9.75617 

9-75  647 
9.75  676 
9-75  705 
9-75  735 
9.75  764 
9-75  793 
9-75  822 

29 
30 
29 
29 
30 
29 
29 
29 

0.24412 
0.24  383 
0-24  353 
0.24  324 
0.24  293 
0.24  265 
0.24  236 
0.24  207 
0.24  178 

9-93  891 
9-93  884 
9-93  876 
9.93  869 
9-93  862 
9-93  855 

9-93  847 
9.93  840 

9-93  833 

I 

7 
7 
7 
8 

7 
7 

19 

18 

i? 
16 
'5 
H 
13 

12 
II 

5  20.6  19.9 
24.4  23.6 
£  28.1  27.2 

7    7 
30   29 

50 

9.69  677 

9.7^  852 

0.24  148 

9-93  826 

10 

°    21    21 

5i 

52 
53 
54 

H 
% 

59 

9.69721 
9-69  743 

9-69  765 
9.69  787 
9.69  809 
9.69831 
9.69  853 
9.69875 

22 

22 
22 
22 
22 
22 
22 
22 

9-75  881 
9.75910 

9-75  939 
9-75  969 
9-75  998 
9.76027 

9.76  056 
9.76086 

"•7"  "5 

29 
29 
30 
29 
29 

30 
29 

0.24  119 
0.24090 
0.24  061 
0.24031 
0.24  002 
0.23973 
0.23  944 
0.23914 

O.2.;  885 

9.93819 
9.93811 
9-93  804 
9-93  797 
9-93  789 
9.93  782 

9-93  775 
9-93  76^ 
9  93  76o 

8 

7 
7 

8 

7 
7 
7 
8 

9 
8 
7 
6 
5 
4 
3 

2 
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1   6.4   6.2 

10.7   10-4 
6   15.0   14.5 

J  19-3  18.6 

1   23.6   22.8 
7   27.9   26.9 

60 

9.69  897 

9.76  144 

0.23  856 

9-93  753 

0 

L.  Cos. 

d. 

L.  Cot. 

c.  d. 

L.  Tan. 

L.  Sin, 

d. 

' 

P.P. 

K'M'I>  SI.-KV.  — 29 


60' 


450 


30' 


L.  Sin. 

d. 

L.  Tan.    c.  d.     L.  Cot. 

L.  Cos.       d. 

p.  p.           , 

o 

9.69897      22 

9.76  144 

20 

0.23856 

9-93  753 

6O 

I 

9-699I9        22 

9-76  173 

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0.23  827 

9-93  746 

59 

2 

9.69941         ,„ 

9.76  202 

29 

0.23  798 

9-93  738 

58 

3 

9.69963        ~2* 

9-76  231 

29 

0.23  769 

9-93  73' 

7 

57 

4 

9-69  984        22 

9.76  261 

3° 

0-23  739 

9-93  724 

7 

56 

30     29      28  i 

5 

9.70006 

9.76  290 

29 

0.23  710 

9.937I7 

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3.0     2.9     2.8 

6 

9.70028        ^ 

9.76319 

29 

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2 

6.0    5.8     5.6 

7 
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9.70050 
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29 
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9.70093        „„ 

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29 

2Q 

0.23  594 

9-93  687 

5i 

5 

15.0    14.5     14.0 

10 

ii 

12 
13 

9.70II5 
9.70  137 
9.70159 
9.70  1  80 

22 
22 
21 

9-76435 
9.76  464 

9-76493 
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29 
29 
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0.23  505 

9.93  680 

7 
8 

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c 

50 

49 
48 

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24.0  23.2  22.4 
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0.23  530 
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9.93  673 
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9.70  202 

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9.70  224 

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22       21 

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0.23  332 

9-93621 

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2.2          2.1 

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9.70310 

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9.76697 

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0.23  303 

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9.76  725 

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0-23  275 

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0.23217 

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9.70396 

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0.23  1  88 

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9.76841 
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0.23  159 
0.23  130 
0.23  101 

9-93577 
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0.8    0.7 

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9-70  590 

9-77073 

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z     1.6     1.4 

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9.70611 

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9-93  495 
9-93  487 

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25 
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0.14087 
0.14060 
0.14033 
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o-I3954 
0.13927 
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9.90  N;S 
9.90  869 
9.90  860 
9.90851 
9.90  842 
9.90  832 
9.90  823 
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9.90  805 

9 
9 
9 
9 

10 

9 
9 
9 

9 

9 
8 
7 
6 
5 
4 
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2 

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2 

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7 
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1.5   1.4 

4-5   4-3 
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16.5  15.9 
19.5  18.8 
22.5  21.7 
25.5  24.6 

60 

9.76  022 

9.86  126 

0.13874 

9.90  796 

0 

L.  Cos, 

d. 

L,  Cot, 

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L.  Tan. 

L.  Sin, 

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L.  Sin. 

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L.  Tan. 

c.  d. 

L.  Cot. 

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d, 

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9.76  922 

17 

9.86  126 

27 

0.13874 

9.90  796 

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2 

3 
4 
5 
6 

7 
8 
9 

9-76  939 
9.76957 
9.76  974 
9.76991 
9.77009 
9.77  026 

9-77  043 
9.77061 
9.77078 

18 
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17 
18 
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18 
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17 

9-86  153 
9.86179 
9.86  206 
9.86  232 
9.86  259 
9.86  285 
9.86312 
9-86  338 
9-86  365 

26 
27 
26 

3 

27 
26 
27 

0.13847 
0.13821 
0.13794 
0.13  768 
0.13  741 
O.I37I5 
0.13688 
0.13  662 
0-13635 

9.90  787 
9.90  777 
9.90  768 

9-90  759 
9.90  750 
9.90  741 
9.90  731 
9.90  722 
9.90713 

10 

9 
9 
9 
9 

10 

9 
9 

59 
58 
57 
56 
55 
54 
53 
52 
51 

I 

2 

3 
4 

27   26 

2.7   2.6 
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8.1   7.8 
10.8  10.4 
13-5  13-0 

10 

9-77095 

17 

9-86  392 

26 

o.  1  3  608 

9.90  704 

50 

6 

1  6.2  15.6 

ii 

12 
13 

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15 
16 

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19 

9.77  112 
9-77  13° 
9-77  M7 
9-77  164 
9-77  181 
9.77  199 
9.77216 
9-77  233 
9-77  250 

18 
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17 
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9.86418 
9-86  445 
9.86471 
9.86  498 
9-86  524 
9-86551 
9.86577 
9.86  603 
9.86  630 

27 
26 

27 
26 
27 
26 
26 

27 
26 

0.13  582 

0-13555 
0.13529 

0.13  502 

0.13476 
0.13449 
0.13423 
0-13397 
0.13370 

9.90  694 
9.90  685 
9.90  676 
9.90  667 
9.90657 
9.90  648 

9-90  639 
9.90  630 
9.90  620 

9 
9 
9 

10 

9 
9 
9 

10 

49 
48 

47 
46 
45 
44 
43 
42 
4i 

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9 

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2 

18.9  18.2 

21.6   20.8 

24.3  23.4 

18   17   16 

1.8  1.7  1.6 
3.6  3.4  3.2 

20 

9.77  268 

17 

9.86  656 

0-13  344 

9.90611 

40 

3 

5-4  5-i  4-8 

21 
22 
23 

24 
25 
26 

3 

29 

9-77  283 
9-77  3°2 
9-773I9 
9-7-7  336 
9-77  353 
9-77  37° 
9-77  387 
9-77  405 
9.77422 

17 
17 

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9.86  683 
9.86  709 
9-86  736 
9.86  762 
9.86  789 
9.86815 
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9.86  868 
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26 
27 
26 

27 
26 

27 
26 
26 

O.I33I7 
0.13  291 
0.13  264 
0.13  238 
0.13211 
0.13  185 
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9.90  602 
9.90  592 
9.90  583 

9-90574 
9.90  565 

9.90555 
9.90  546 

9.90537 
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10 

9 
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39 
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36 
35 
34 
33 
32 
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4 

6  i 

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26 

0.13079 

9.90518 

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10   9 

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32 
33 
34 
35 
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39 

9-77  456 
9-77473 
9-77  490 
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9-77  524 
9-77541 
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9-77  592 

17 

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17 
17 
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9-86  947 
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9.87027 
9.87053 
9.87  079 

9.87  106 
9-87  132 
9.87158 

27 
26 

27 
26 
26 

27 
26 
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0.13053 
0.13026 
0.13000 
0.12973 
0.12947 
0.12921 

0.  1  2  894 
0.12868 
0.12842 

9.90  509 
9.90499 
9.90490 
9.90  480 
9.90471 
9.90  462 
9.90452 
9.90  443 
9-90  434 

10 

9 

10 

9 
9 
10 

9 
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29 
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27 
26 
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23 

22 
21 

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2 

3 
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9 

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2.0   1.8 

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7-0  6.3 
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40 

9-77  6o9 

9.87  1  8? 

26 

0.12  815 

9.90  424 

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9.77  626 
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9.87  211 
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27 
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0.12789 
0.12  762 

9.90415 
9.90  405 

10 

19 

18 

43 
44 
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3 

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9.77  660 
9.77677 

9-77  694 
9.77711 

9-77  728 
9-77  744 
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9.87  264 
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9-87  343 

9-87  369 
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26 

27 
26 
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3 

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0.12736 
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0.12683 
O.I2  657 
O.I2  631 
0.  1  2  604 
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9.90  396 
9.90  386 
9-90  377 
9.90368 

9.90358 
9-90  349 
9-90  339 

10 

9 
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10 

9 

10 

9 

17 
16 
15 
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12 
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27   26 
1.5   1.4 

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9.77778 

17 

9.87  448 

27 

0.12552 

9-90  330 

10 

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2 

4-5   4-3 

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52 
53 
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11 

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9-77  795 
9.77812 
9-77  829 
9-77846 
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9-77  879 
9.77  896 
9-779I3 
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16 
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9.87475 
9.87  501 
9.87527 

9-87554 
9-87  580 
9.87  606 

9-87  633 
9.87  659 
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26 
26 

27 
26 
26 

27 
26 
26 
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0.12525 
0.12499 
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9.90  320 
9.90311 
9.90  301 
9.90  292 
9.90  282 
9.90  273 
9.90  263 
9.90  254 
9.90  244 

9 

10 

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10 

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7 
6 
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4 
3 

2 

3 
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6 

9 

10.5  10.1 
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60 

9.77946 

9.87711 

0.12  289 

9.90  235 

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L.  Cot. 

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53° 


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L.  Sin. 

d. 

L.  Tan. 

c.  d. 

L.  Cot. 

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d. 

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9-77  946 

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0.  1  2  289 

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9-77  963 
9.77980 

9-77  997 
9.78013 
9.78  030 
9.78047 
9.78  063 
9.78  080 
9.78097 

17 
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16 

17 
17 
16 

16 

9.87  738 
9-87  764 
9.87  790 

9.87817 
9.87  843 
9.87  869 
9.87  895 
9.87  922 
9.87  948 

26 

26 

27 
26 
26 
26 

27 
26 

0.12  262 
0.12  236 
0.12  210 
O.I2  183 
0.12  157 
0.12  131 
O.I  2  lO^ 
O.I2O78 
0.12052 

9.90  225 
9.90216 
9.90  206 
9.90  197 
9.90  187 
9.90178 
9.90  168 
9.90159 
9.90  149 

9 
10 

9 

IO 

9 

IO 

9 

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3 

57 
56 
55 
54 
53 
52 
51 

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2 

3 
4 

5 

27   26 

2.7   2.6 

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10.8  10.4 

13-5   !3-0 

10 

9.78  113 

9.87  974 

26 

0.12026 

9.90  139 

SO 

ii 

12 
13 

ii 
;; 

19 

9.78  130 
9.78  147 
9.78  163 
9.78  1  80 
9.78  197 
9.78213 

9-78  230 
9.78  246 
9.78  263 

17 
16 

17 
*7 
16 

16 
17 

9.88000 
9.88027 
9.88053 
9.88079 
9.88  105 
9.88  131 
9.88  158 
9.88  184 
9.88210 

27 
26 
26 
26 

26 

26 
26 

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O.I2OOO 
O.II973 
0.11947 

o.u  921 
o.u  895 
o.u  869 
o.u  842 
o.u  816 
o.u  790 

9.90  130 
9.90  1  20 
9.90  1  1  1 
9.90  101 
9.90091 
9.90  082 
9.90  072 
9.90063 
9.90053 

10 

9 

10 
IO 

9 

10 

9 

10 

49 
48 

47 
46 
45 
44 

43 
42 

8 
9 

i 

2 

21.6   20.8 

24-3  23.4 

17   16 
1.7   1.6 
3-4   3-2 

20 

9.78  280 

16 

9.88  236 

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o.u  704 

9.90  043 

40 

3 

5.1   4.8 

21 
22 
23 
24 
25 
26 

27 

28 
29 

9.78  296 

9-78313 
9.78329 

9.78  346 
9.78362 
9-78  379 

9-78  395 
9.78412 
9.78428 

17 
16 

17 
16 

17 
16 

16 

9.88  262 
9.88  289 
9-88315 
9.88  341 
9-88  367 
9-88  393 
9.88  420 
9.88  446 
9.88472 

3 

26 
26 
26 

26 
26 
26 

o.u  738 
o.u  711 
o.u  685 
o.u  659 

0.11633 

o.u  607 
o.u  580 

0.11554 

o.u  528 

9.90  034 
9.90  024 
9.90014 
9.90  005 
9-89  995 
9-89  985 
9-89  976 
9.89  966 
9.89  956 

IO 

10 

9 

IO 
10 

9 

IO 
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37 
36 
35 
34 

3i 

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9 

6.8   6.4 
8.5   8.0 

10.2    9.6 
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13.6  12.8 

15.3   14.4 

30 

9-78445 

16 

9.88  498 

26 

o.i  i  502 

9.89  947 

3O 

10   9 

31 
32 

33 
34 
35 
36 

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39 

9.78461 
9.78478 
9.78  494 
9.78510 
9-78527 
9-78  543 
9.78  560 
9.78576 
9.78592 

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16 

17 

16 

16 
16 

9-88  524 
9-88  550 
9-88  577 
9.88  603 
9.88  629 
9.88  655 
9.88681 
9.88  707 
9-88  733 

26 
27 
26 
26 
26 
26 
26 
26 
26 

o.u  476 
o.  1  1  450 

0.11423 

o.i  i  397 
0.11371 
0.11345 

0.11319 
o.u  293 
o.u  267 

9-89  937 
9.89927 
9.89918 
9.89  908 
9.89  898 
9.89  888 
9.89  879 
9.89  869 
9.89  859 

10 

9 

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10 
10 

9 

10 
10 

29 
28 
27 
26 
25 
24 
23 

22 
21 

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2 

3 
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1.0  0.9 

2.0   1.8 

3.0  2.7 

4-0  3-6 

5-°  4-5 
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40 

9.78609 

16 

9-88  759 

o.u  241 

9-89  849 

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9.78625 
9  78  642 

9.88  786 
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26 

o.u  214 
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9.89  840 
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19 

18 

43 

44 
45 
46 

49 

9.78  658 

9-78  674 
9.78691 
9.78  707 

9-78  723 
9-78  739 
9.78  756 

16 
16 

17 
16 
16 
16 

17 
16 

9.88  838 
9.88  864 
9.88  890 
9.88916 
9.88  942 
Q.88  968 
9.88  994 

26 
26 
26 
26 
26 
26 
26 
26 

o.u  162 
o.u  136 
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o.u  084 
o.u  058 
o.u  032 

0.  1  I  OO6 

9.89  820 
9.89  810 
9.89  So  i 
9.89  791 
9.89  781 
9.89  771 
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10 
10 

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10 
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17 
16 

14 
13 

12 
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10   10 
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9.78  772 

16 

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26 

0.10980 

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6.8   6.5 

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9.78  886 
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16 
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9.89  046 
9.89  073 
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9.89  125 
9.89151 
9.89177 
9.89  203 
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26 
26 
26 
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0.10954 
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o.io  771 

0.10745 

9.89  742 
9.89  732 
9.89  722 
9.89712 
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9.89  307 
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9-89  463 
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26 
26 
26 
26 
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9-89  584 
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56 
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54 
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9.89  554 

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7  18.2  17-5 

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9.79111 
9.79  128 
9-79  144 
9.79  160 
9.79  176 
9-79  192 
9.79  208 
9-79  224 
9-79  240 

17 
16 
16 
16 
16 
16 
16 
16 
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9.89  567 

9-89  593 
9.89  619 

9.89  645 
9.89671 
9.89  697 
9.89  723 
9-89  749 
9.89  775 

26 
26 
26 
26 
26 
26 
26 
26 
26 

0.10433 

0.10407 
0.10381 

0.10355 

o.io  329 
0.10303 
o.io  277 
o.io  251 
o.io  225" 

9.89  544 
9-89  534 
9.89  524 

9.89514 
9.89  504 
9-89  495 
9-89  485 
9-89  475 
9-89  465 

10 
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10 
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10 
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49 
48 
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46 
45 
44 
43 
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8   2O.8   2O.O 

9  23-4  22.5 

17   16   15 

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2   3.4  3.2  3.0 

20 

9-79  256 

16 

9.89801 

26 

o.io  199 

9-89  455 

IO 

40 

4   68  64  60 

21 
22 
23 

24 

25 
26 

27 
28 
29 

9.79  272 
9.79  288 
9-79  304 
9-793I9 
9-79  335 
9-79  35  ! 
9-79  367 
9-79  383 
9-79  399 

16 
16 

15 
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16 
16 
16 
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9.89  827 
9-89  853 
9.89  879 

9-89  905 
9.89931 

9-89957 

9-89  983 
9.90  009 
9-90035 

26 
26 

26 

26 
26 
26 
26 

o.io  173 
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O.IO  121 

o.io  095 
0.10069 
0.10043 
0.10017 
0.09  991 
0.09  965 

9.89  445 
9-89  435 
9.89425 
9.89415 
9.89  405 
9-89  395 
9-89  385 
9-89  375 
9.89  364 

10 
10 
10 
10 
10 
10 
10 

II 

39 
38 
37 
36 
35 
34 
33 
32 

5   8.5  8.0  7.5 
6  10.2  9.6  9.0 
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8  13.6  12.8  12.0 
9  15-3  14-4  13-5 

H  10  9 

30 

9-794I5 

16 

9.90061 

0.09  939 

9-89  354 

30 

32 
33 
34 
35 
36 

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39 

9-79  431 

9-79  447 
9-79  463 
9-79478 
9-79  494 
9.79510 

9.79  526 
9-79  542 
9-79  55s 

16 
16 

16 
16 
16 
16 
16 

9.90  080 

9.90  I  I  2 
9.90  138 
9.90  I64 
9.90  190 

9.90  216 

9.90  242 

9.90  268 

9.90  294 

25 
26 
26 
26 
26 
26 
26 
26 
26 
26 

0.09914 
0.09  888 
0.09  862 
0.09  836 
0.09  810 
0.09  784 
0.09  758 
0.09  732 
0.09  706 

9-89  344 
9-89  334 
9.89  324 

9.89  314 
9.89  304 
9.89  294 
9.89  284 
9.89  274 
9.89  264 

10 

10 
10 
10 
IO 
10 
10 
10 

29 
28 
27 
26 
25 
24 
23 

22 
21 

i  i.i  i.o  0.9 

2   2.2  2.0  1.8 

3  3-3  3-o  2,7 
4  4.4  4.0  3.6 

5  5-5  5-°  4-5 
6  6.6  6.0  5.4 
7  7-7  7-o  6.3 
8  8.8  8.0  7.2 
9  9.9  9.0  8.1 

40 

9-79  573 

16 

9.90  320 

26 

0.09  680 

9-89  254 

20 

42 

9-79  589 
9.79  605 

16 

9.90  346 
9.90371 

0.09  654 
0.09  629 

9.89  244 
9  89  233 

II 

19 
18 

43 

44 

3 

:? 

49 

9.79621 

9-79  636 
9.79652 
9.79668 

9-79  684 
9.79  699 
9-79  71  5 

16 
16 
16 

16 

9-90  397 
9.90423 
9.90  449 
9.90  475 
9.90  501 
9.90527 
9-90  553 

26 
26 
26 
26 
26 

0.09  603 
0.09  577 
0.09551 
0.09  525 
0.09  499 
0.09  473 
0.09  447 

9.89  223 
9.89213 
9.89  203 
9.89  193 
9.89  183 
9.89173 
9.89  162 

IO 
10 
10 
10 
10 

II 

*7 

16 
15 
H 
'3 

12 
II 

10   10   9 
26   25   26 
0  1.3  1.2  1.4 

50 

9-79  73i 

9.90578 

26 

0.09  422 

9.89  152 

10 

2  j-9  3-°  4-3 

65  62  72 

52 
53 
54 

3 

5* 
59 

9-79  746 
9-79  762 
9-79  778 
9-79  793 
9-79  809 
9-79  825 
9-79  840 
9-79  856 
9-79  872 

'5 

16 
16 

16 
16 

16 
16 

9.90  604 
9.90  630 
9.90  656 
9.90  682 
9.90  708 
9-90  734 

9-90  759 
9.90  785 
9.90  8  1  1 

26 
26 
26 
26 
26 

26 
26 
26 

0.09  396 
0.09  370 
0.09  344 
0.09318 
0.09  292 
0.09  266 
0.09  241 
0.09  215 
0.09  189 

9.89  142 
9.89  132 

9.89  122 
9.89  112 

9.89  ioi 

9.89091 
9.89081 
9.89071 

9.89  odo 

IO 
10 
10 

10 

IO 
10 

II 

9 

8 

7 
6 

4 
3 

2 

3  9.1  8.8  10.1 
4!  11.7  11.  2  13.0 
g  14.3  13-8  15-9 
16.9  16.2  18.8 
7  19.5  1  8.8  21.7 

22.1  21.2  24.6 
tj  24.7  23.8  - 

60 

9-79  887 

9.90  837 

0.09  163 

9.89  050 

0 

L.  Cos. 

d. 

L.  Cot. 

c.  d. 

L.  Tan. 

L.  Sin. 

d. 

' 

P.P. 

51° 


39' 


459 


' 

L.  Sin. 

d. 

L.  Tan. 

c.  d. 

L.  Cot. 

L.  Cos. 

d. 

P.P. 

o 

9.79  887 

16 

9.90  837 

26 

0.09  163 

9.89  050 

6O 

2 

3 

4 
5 
6 

9 

9-79  903 
9.79918 
9-79  934 
9-79  95° 
9-79  965 
9-79  98i 
9.79  996 
9.80012 
9.80027 

15 

16 
16 

!i 

15 
16 

15 
16 

9.90  863 
9.90  889 
9.90914 
9.90  940 
9.90  966 
9.90  992 
9.91  018 
9.91  043 
9.91  069 

26 
25 
26 
26 
26 
26 

25 
26 
"6 

0.09  137 
0.09  1  1  1 
0.09  086 
0.09  060 
0.09  034 
0.09  008 
0.08  982 
0.08957 
0.08931 

9.89  040 
9.89  030 

9.89  020 
9.89  009 

9.88  999 
9.88  989 
9.88  978 
9.88  968 
9.88958 

10 
IO 

II 

10 
10 

II 

10 
10 
IO 

59 

58 
57 
56 
55 
54 
53 
52 
5i 

2 

3 
4 

I 

26   25 

2.6   2.5 
5.2   5.0 
7-8   7-5 
10.4  10.0 
13.0  12.5 
ic  6  no 

10 

9.80  043 

9.91  095 

26 

0.08  905 

9.88  948 

5O 

7  : 

18.2  17.5 

ii 

12 
13 

14 

15 

3 

19 

9.80  058 
9.80  074 
9.80  089 
9.80  105 

9.80  120 
9.80  136 
9.80I5I 

9.80  1  66 
9.80  182 

16 
15 
16 

II 

15 

15 
16 
i  ^ 

9.91  121 

9-91  H7 
9.91  172 

9.91  198 
9.91  224 
9.91  250 
9.91  276 
9.91  301 
9.91  327 

26 
25 
26 
26 
26 
26 

25 
26 

"6 

0.08  879 
0.08  853 
0.08  828 
0.08  802 
0.08  776 
0.08  750 
0.08  724 
0.08  699 
0.08  673 

9.88937 
9.88  927 
9.88917 
9.88  906 
9.88  896 
9.88  886 
9.88  875 
9.88  865 
9.88855 

IO 
IO 

II 

10 
10 

II 

IO 
IO 
1  1 

49 
48 

47 
46 
45 
44 

43 
42 

4i 

8 
9 

2 

3 

20.8   20.0 

23.4  22.5 

16   15 

1.6   1.5 
3-2   3-o 
4-8   4-5 

2O 

9.80  197 

16 

9-9i  353 

"6 

0.08  647 

9.88  844 

IO 

4O 

4 

6.4   6.0 

21 
22 
23 
24 
25 
26 

3 

29 

9.80213 
9.80  228 
9.80  244 
9.80  259 
9.80  274 
9.80  290 
9.80  305 
9.80  320 
9.80  336 

15 
16 

1S 
15 
16 

15 

:i 

9-9i  379 
9.91  404 
9.91  430 
9.91  456 
9.91  482 
9.91  507 

9-9i  533 
9-9i  559 
9-91  585" 

25 
26 
26 
26 
25 
26 
26 
26 

0.08  621 
0.08  596 
0.08  570 
0.08  544 
0.08518 
0.08  493 
0.08  467 
0.08441 
0.08415 

9.88834 
9.88  824 
9.88813 
9.88  803 
9-88  793 
9.88  782 
9.88  772 
9.88  761 
9.88751 

IO 

II 

10 
10 

II 

10 

II 

10 

10 

39 
38 
37 
36 
35 
34 
33 
32 
31 

5 
6 

I 

9 

8.0   7.5 
9.6   9.0 

II.  2   10.5 
12.8   12.0 
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11   10 

30 

9.80351 

9.91  610 

26 

0.08  390 

9.88  741 

3O 

3i 

32 
33 
34 
35 
36 

37 
38 
39 

9.80  366 
9.80  382 
9.80  397 
9.80412 
9.80  428 
9.80  443 
9.80458 
9.80  473 
9.80  489 

16 

!5 

IS 
16 
15 
15 

!i 

9.91  636 
9.91  662 
9.91  688 

9-9i  713 
9-9i  739 
9.91  765 

9.91  791 
9.91  816 
9.91  842 

26 
26 

25 
26 
26 
26 

25 
26 
26 

0.08  364 
0.08  338 
0.08  3  1  2 
0.08  187 
0.08  261 
0.08  235 
0.08  209 
0.08  184 
0.08  158 

9-88  730 
9.88  720 
9.88  709 
9.88  699 
9.88  688 
9.88  678 
9.88  668 
9-88  657 
9.88  647 

10 

II 

IO 

II 

10 
10 

II 

IO 

29 
28 
27 
26 
25 
24 
23 

22 
21 

3 
4 

I 

9 

3-3  3-o 
4.4  4.0 

1:1  IS 

7-7  7-° 
8.8  8.0 

9-9  9-0 

4O 

9.80  504 

I  c 

9.91  868 

2; 

0.08  132 

9.88  636 

IO 

20 

4i 
42 

43 
44 
45 
46 

47 
48 

49 

9.80519 
9.80  534 
9.80550 
9-80  565 
9.80  580 
9-8o  595 
9.80610 
9.80  625 
9.80  641 

15 
16 

J5 
15 
X5 
15 

;i 

9.91  893 
9.91919 
9-91  945 
9.91  971 
9.91  996 
9.92022 
9.92  048 
9.92073 
9.92  099 

26 
26 
26 

25 
26 
26 

% 

26 

0.08  107 
0.08081 
0.08  05  5 
0.08  029 
0.08  004 
0.07  978 
0.07  952 
0.07  927 
0.07  901 

9.88  626 
9.88615 
9.88  605 
9.88  594 
9.88  584 
9.88  573 
9.88  563 
9.88552 
o.SX  -,42 

II 

10 

II 

IO 

II 

10 

10 

19 
18 
»7 

16 
15 
H 
13 

12 
II 

0 

i 

11   11 

26   25 

1.2    I.I 

3-5   3-4 

5-9   5-7 

5O 

9.80  656 

9.92125 

0.07  875 

9.88531 

IO 

3 

8.3   8.0 

51 

52 
53 
54 

II 

57 
5» 
59 

9.80671 
9.80  686 
9.80  701 
9.80  716 
9.80  731 
9.80  746 
9.80  762 
9.80  777 
9.80  792 

15 
15 

'5 
1S 

'5 
16 

i5 
15 

9.92  150 
9.92  176 

9.92  202 
9.92  227 
9.92  253 
9.92  279 
9.92  304 
9-92  330 
9.92  356 

25 

26 
26 

25 
26 
26 

25 
26 
26 

0.07  S^o 
0.07  S.?4 
0.07  798 
0.07  773 
0.07  747 
0.07  721 
0.07  696 
0.07  670 
0.07  644 

9.88  =;-•! 
9.88510 
9.88  499 
9.88  489 

9.SS  478 
y.SS  408 

9.88457 

9.88  447 
9.88  436 

1  1 
1  1 

10 

II 

10 

II 

IO 

II 
1  1 

9 
8 
7 
6 
5 
4 
3 

2 

9 

10 

ii 

IO.6   IO.2 

13.0  12.5 

15.4   14.  S 

20.1    19.3 
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24.8   23.9 

60 

9.80807 

9.92  38l 

0.07  619 

9.88425 

0 

L_ 

L.  Cos. 

d. 

L,  Cot, 

c.  d. 

L.  Tan. 

L.  Sin. 

d. 

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P.P. 

50' 


460 


40' 


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L.  Sin. 

d. 

L.  Tan. 

c.  d. 

L.  Cot. 

L.  Cos. 

d. 

P.P.        I 

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9.80807 

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9.92  381 

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0.07  619 

9-88  425 

IO 

60 

I 

2 

3 

4 

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9 

9.80  822 
9.80837 
9.80  852 
9.80  867 
9.80  882 
9.80  897 
9.80912 
9.80927 
9.80  942 

15 
15 
15 
15 
15 
15 

'5 

9.92  407 
9-92  433 
9.92  458 

9.92  484 
9.92510 
9-92  535 
9.92561 
9.92  587 
9.92  612 

26 

25 
26 
26 
25 
26 
26 

3 

0-07  593 
0.07  567 
0.07  542 
0.07  516 
0.07  490 
0.07  465 
0.07  439 
0.07413 
0.07  388 

9.88415 
9.88  404 
9-88  394 
9-88  383 
9-88  372 
9.88  362 
9.88351 
9.88  340 
9-88  330 

ii 

IO 

II 
II 

IO 

II 
II 

10 

P 

57 
56 
55 
54 
53 
52 

2 

3 
4 
5 
6 

26   25 

2.6   2.5 
5-2   5-° 
7-8   7-5 
10.4  i  o.o 

13.0  12.5 
15.6  15.0 

10 

9.80957 

9.92  638 

0.07  362 

9.88319 

50 

7 
g 

18.2  17.5 

ii 

12 
13 

IS 

3 

19 

9.80  972 
9.80  987 

9.81  002 
9.8l  017 
9.8l  032 
9.8l  047 

9.81  061 
9.81  076 
9.81  091 

15 
15 
15 
15 
15 

'5 

15 

9.92  663 
9.92  689 
9.92715 
9.92  740 
9.92  766 
9.92  792 
9.92817 
9.92  843 
9.92  868 

26 
26 

25 
26 
26 

25 
26 

3 

0-07  337 
0.07  311 
0.07  285 
0.07  260 
0.07  234 
0.07  208 
0.07  183 
0.07  157 
0.07  132 

9.88  308 
9.88  298 
9.88  287 
9.88  276 
9.88  266 
9-88  255 
9.88  244 
9-88  234 
9.88  223 

10 

II 
II 

IO 

II 

II 

IO 

II 

49 
48 
47 
46 
45 
44 
43 
42 

9 

i 

2 

3 

23.4  22.5 

15   14 
1.5   1.4 

3.0   2.8 
4-5   4-2 
60   56 

20 

9.81  106 

1C 

9.92  894 

26 

0.07  1  06 

9.88212 

40 

7-5   7-° 

21 
22 
23 
24 

3 

27 
28 
29 

9.81  121 
9.81  136 
9.81  151 
9.81  166 
9.81  1  80 
9.81  195 

9.8l  210 
9.8l  225 
9.8l  240 

15 
15 
15 

H 

15 
15 

9.92  920 
9.92  945 
9.92971 
9.92  996 
9-93  022 
9-93  048 
9-93  073 
9-93  099 
9.93124 

25 
26 

25 
26 
26 

25 
26 

25 

0.07  080 
0.07  055 
0.07  029 
0.07  004 
0.06  978 
0.06952 
0.06  927 
0.06  901 
0.06  876 

9.88  201 
9.88  191 
9.88  1  80 
9.88  169 
9.88  158 
9.88  148 

9-88  137 
9.88  126 
9.88115 

IO 

II 
II 
II 

10 

II 
II 

II 

39 
38 
37 
36 
35 
34 
33 
32 

7 
8 

9 

i 

9.0   8.4 
10.5   9.8 

I2.O   II.  2 

13.5  12.6 
11   10 

I.I    1.0 

30 

9.8l  254 

9.93  150 

0.06  850 

9.88  105 

30 

2 

2.2    2.0 

32 
33 
34 
35 
36 

3 

39 

9.8l  269 
9.8l  284 
9.8l  299 
9.8l  314 
9.8l  328 

9-8  1  343 
9.81  358 
9.81  372 
9.81  387 

15 
15 
15 
H 
15 
15 

15 

IT 

9-93  175 
9-93  201 
9-93  227 
9-93  252 
9-93  278 
9-93  303 
9-93  329 
9-93  354 
9-93  38o 

26 
26 

25 
26 
25 
26 

n 

26 

0.06  825 
0.06  799 
0.06  773 
0.06  748 
0.06  722 
0.06  697 
0.06671 
0.06  646 
0.06  620 

9.88  094 
9.88083 
9.88  072 
9.88061 
9.88051 
9.88  040 
9.88029 
9.88018 
9.88  007 

II 
II 
II 

IO 

II 
II 
II 
II 

II 

29 
28 

27 
26 
25 
24 
23 

22 
21 

3 

4 

I 
I 

9 

3-3   3-o 

44   4.0 

5-5   5-o 
6.6   6.0 

7-7   7-° 
8.8   8.0 
9-9   9-o 

40 

9.81  402 

9.93  406 

0.06  594 

9.87  996 

20 

41 
42 
43 
44 

47 
48 
49 

9.81  417 
9.81  431 
9.81  446 
9.81  461 
9-8i  475 
9.81  490 
9.81  505 
9.81  519 
9-8  1  534 

14 
15 

'5 

14 
15 
15 

15 

1C 

9-93431 
9-93457 
9-93  482 
9-93  5°8 
9-93  533 
9-93  559 

9-93  584 
9.93610 
9-93  636 

25 
26 
25 
26 

3 

25 
26 
26 

0.06  569 
0.06  543 
0.06  518 
0.06  492 
0.06  467 
0.06  441 
0.06416 
0.06  390 
0.06  364 

9.87  985 
9-87  975 
9.87  964 

9-87  953 
9.87  942 

9-87931 
9.87  920 
9.87  909 
9.87  898 

IO 

II 
II 
II 
II 
II 
II 
II 
II 

19 
18 

17 
16 
'5 
14 
13 

12 
II 

0 
2 

3 

11   10   10 
26  26  25 

1.2   1.3   1.2 

3-5  3-9  3-8 
5-9  6.5  6.2 

SO 

9.81  549 

14 

9-93  66  1 

26 

0.06  339 

9.87  887 

IO 

10 

4 

8.3  9.1  8.8 

52 
53 
54 
55 
56 

H 

59 

9.81  563 
9.81  578 
9.81  592 
9.81  607 
9.81  622 
9.81  636 
9.81  651 
9.81  665 
9.81  680 

15 
15 

15 

IS 
14 

9-93  687 
9-93  712 
9-93  738 
9-93  763 
9-93  789 
9.93814 

9.93  840 
9-93  865 
9-93  891 

25 
26 
25 
26 

25 
26 

0.06313 
0.06  288 
0.06  262 
0.06  237 

O.06  211 

0.06  1  86 
0.06  1  60 
0.06  135 
0.06  109 

9.87  877 
9.87  866 
9-87  855 
9.87  844 
9-87  833 
9.87  822 

9.87811 
9.87  800 
9.87  789 

II 
II 
II 
II 
II 
II 
II 
II 

1  1 

9 
8 

7 
6 
5 
4 
3 

2 

I 

«  O  \O  00-vJ  ON<-" 

3.0  14.3  13.8 
5.4  16.9  16.2 
7.7  19.5  18.8 

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60 

9.81  694 

9.93916 

0.06  084 

9.87  778 

0 

L.  Cos. 

d. 

L.  Cot. 

c.  d. 

L.  Tan. 

L.  Sin. 

d. 

t 

P.  P. 

49C 


41° 


461 


,  / 

L.  Sin. 

d. 

L.  Tan. 

c.  d. 

L.  Cot. 

L.  Cos. 

d. 

p.  P. 

o 

9.81  694 

9.93916 

26 

0.06  084 

9-87  778 

6O 

I 

2 

3 

4 

I 

7 

8 

9 

9.81  709 
9.81  723 
9.81  738 
9.81  752 
9.81  767 
9.81  781 
9.81  796 
9.81  810 
9.81  825 

>4 
15 
H 
15 

H 
15 

9-93  942 
9-93  967 
9-93  993 
9.94018 
9.94  044 
9.94  069 

9-94  095 
9.94120 
9.94  146 

25 
26 

25 
26 
25 
26 

3 

2s 

0.06  058 
0.06  033 
0.06  007 
0.05  982 
0.05  956 
0.05931 
0.05  905 
0.05  880 
0.05  854 

9.87  767 
9.87  756 
9-87  745 

9-8?  734 
9.87  723 
9.87712 
9.87  701 
9.87  690 
9-87  679 

ii 
ii 
ii 
ii 
n 
ii 
ii 
1  1 
1  1 

59 
58 
57 
56 
55 
54 
53 
52 
5' 

2 

3 
4 

26   25 

2.6   2.5 
5.2   5.0 
7-8   7-5 
10.4  10.0 
13.0  12.5 
15.6  15.0 

IO 

9.81  839 

9.94171 

26 

0.05  829 

9.87  bbS 

50 

7 

18.2  17.5 

1  1 

12 
13 

14 
15 

17 
19 

9.81  854 
9.81  868 
9.81  882 
9.81  897 
9.81  911 
9.81  926 
9.81  940 

9.81955 
9.81  969 

15 
'4 
'5 

14 

14 

9-94  197 

9-94  222 
9-94  248 

9-94  273 

9.94  299 

9-94  324 
9-94  35° 
9-94  375 
9.94  401 

25 
26 

25 
26 
25 
26 

25 
26 

0.05  803 
0.05  778 
0.05  752 
0.05  727 
0.05  701 
0.05  676 
0.05  650 
0.05  625 
0-05  599 

9-87  657 
9.87  646 

9.87  635 

9.87  624 

9.87613 

9.87  601 
9.87  590 
9-87  579 
9.87  568 

n 
ii 
ii 
ii 

12 
II 
1  1 
II 

49 
48 
47 
46 
45 
44 
43 
42 
41 

8 
9 

I 

2 

3 

20.8   20.0 
23.4   22-5 

15   14 

1.5   1.4 
3.0   2.8 
4-5   4-2 

20 

9.81  983 

9-94  426 

26 

0.05  574 

9-87  557 

II 

4O 

4 

6.0   5.6 

21 

22 
23 
i  24 

••  25 
26 

3 

29 

9.81  998 
9.82012 
9.82  026 
9.82041 
9-82  055 
9.82  069 
9  82  084 
9.82  098 

9.82  112 

14 

15 

14 

'5 
H 
14 

9-94452 
9-94477 
9-94  5°3 
9-94  528 
9-94  554 
9-94  579 
9-94  604 
9-94  630 
9-94  655 

25 
26 

25 
26 
25 

25 
26 

25 

0.05  548 

0-05  523 
0.05  497 

0.05  472 
0.05  446 
0.05  421 
0.05  396 
0.05  370 
0-05  345 

9.87  546 
9-87  535 
9.87  524 

9-87  5J3 
9.87  501 
9.87  490 
9.87  479 
9.87  468 
9-87457 

II 
11 
II 
12 
II 
II 
II 
II 

39 
38 
37 
36 
35 
34 
33 
32 

I 

9 

7-5   7-° 
9.0   8.4 
10.5   9.8 

12.0   I  1.2 

13.5  12.6 
12   11 

3O 

9.82  126 

9.94681 

0.05  319 

9.87  446 

30 

j 

1.2   I.I 

32 
33 

34 

P 

IS 

39 

9.82  141 

9.82  153 
9.82  169 
9.82  184 
9.82  198 

9.82  212 
9.82  226 
9.82  240 
9.82  255 

'4 
H 
15 
14 

H 
15 

9.94  706 
9-94  732 
9-94  757 

9-94  783 
9.94  808 
9.94  834 
9.94  859 
9-94  884 
9.94910 

26 
25 
26 

3 

25 
25 
26 

2C 

0.05  294 
0.05  268 
0.05  243 
0.05  217 
0.05  192 
0.05  1  66 
0.05  141 
0.05  116 
0.05  090 

9-87  434 
9-87423 
9.87412 

9.87  401 
9.87  390 
9-87  378 
9.87  367 
9-87  356 
9-87  345 

II 

II 
II 
II 

12 
II 
I  I 
II 
1  1 

29 
28 
27 
26 
25 
24 
23 

22 
21 

2 

4 
5 
6 

7 
8 
9 

2.4   2.2 

3-6  3-3 
4-8  4-4 
6.0  5.5 
7.2  6.6 

8-4  7-7 
9.6  8.8 
10.8  9.9 

4O 

9.82  269 

14 

9-94  935 

?6 

0.05  065 

9-87  334 

12 

20 

42 
43 
44 
45 
46 

47 
48 
49 

9.82  283 
9.82  297 
9.823II 
9.82  326 
9.82  340 

9-82  354 
9.82  368 
9.82  382 
9.82  396 

»4 

15 

14 

14 

9.94961 
9.94  986 
9.95012 

9-95  °37 
9-95  °62 
9.95  088 

9-95  "3 
9-95  »39 
9-95  l64 

25 

3 

25 
26 

25 
26 

0.05  039 
0.05  014 
0.04  988 
0.04963 
0.04  938 
0.04912 
0.04  887 
0.04  861 
0.04  836 

9.87  322 

9-87311 
9.87  300 

9.87  288 
9.87  277 
9.87  266 

9-87  255 
9.87  243 
9.87  232 

II 
II 
12 
II 
II 
II 
12 
II 
II 

in 

'7 
16 
15 
14 
13 

12 
II 

o 

2 

3 

12  12  11 
26  25  25 

I.I   I.O   I.I 

3-2  3-i  3-4 
5-4  5-2  5-7 
7-6  7-3  8.0 

50 

9.82410 

14 

9-95  !9° 

0.04810 

9.87  221 

12 

IO 

e 

9.8   9.4  IO.2 

51 
52 
53 
54 
55 
56 

P 

59 

9.82  424 
9.82439 
9-82453 
9.82  467 
9.82481 
9.82  493 
9.82  509 
9.82  523 
9-82537 

15 
H 
H 

H 
14 

9-952I5 
9.95  240 
9-95  266 
9-95  291 
9-953I7 
9-95  342 
9-95  368 
9-95  393 
9.95  418 

3 

25 
26 
25 
26 

25 

0.04  783 
0.04  760 
0.04  734 
0.04  709 
0.04  683 
0.04  658 
0.04  632 
0.04  607 
0.04  582 

9.87  209 
9.87  198 
9.87  187 

9-87  *75 
9.87  164 

9-87  153 
9.87  141 
9.87  130 
9.87119 

II 
II 
12 
II 
11 
12 
II 
II 
I  2 

7 
6 
5 
4 
3 

2 

• 

10  \ 

:;> 

1-9  II.3  12.5 
4.1  13.5  14.8 

6.2  15.6  17.0 

8-4  «7-7  '9-3 
0.6  19.8  21.6 
2.8  21.9  23.9 
4.9  24.0  - 

60 

9-82551 

9-95  444 

0.04  556 

9.87  I07 

0 

L.  Cos. 

d. 

L.  Cot. 

c.  d. 

L.  Tan. 

L.  Sin. 

d. 

1 

P.  P. 

48° 


462 


42° 


1 

L.  Sin. 

d. 

L.  Tan. 

c.  d. 

L.  Cot. 

L.  Cos. 

d. 

p.p. 

o 

9-82551 

9-95  444 

0.04  556 

9.87  107 

1  1 

60 

2 

3 
4 

I 

9 

9.82  565 
9.82  579 
9.82  593 
9.82  607 
9.82  621 
9.82  635 
9.82  649 
9.82  663 
9.82  677 

H 
14 
H 
»4 

H 
14 
H 
'4 

9-95  4t>9 
9-95  495 
9-95  520 
9-95  545 
9-95  57i 
9-95  596 
9.95  622 
9-95  647 
9-95  672 

26 
25 
25 
26 

25 
26 

25 
25 

0.04  531 
0.04  505 
0.04  480 
0.04  455 
0.04  429 
0.04  404 
0.04  378 
0.04  353 
0.04  328 

9.87  090 
9.87  085 
9-87073 
9.87  062 
9.87050 
9.87  039 
9.87  028 
9.87016 
9.87  005 

ii 

12 
II 
12 
II 
II 
12 
II 

59 
58 
57 
56 
55 
54 
53 
52 
51 

I 

2 

3 

4 

26   25 

2.6   2.5 
S-2   5-o 
7-8   7-5 
10.4  10.0 

13.0  12.5 

15.6  15.0 

10 

9.82091 

9-95  698 

0.04  302 

9.86  993 

5O 

7 

18.2  17.5 

ii 

12 
13 
14 
15 

16 

17 
18 

19 

9.82  705 
9.82719 
9-82  733 
9.82  747 
9.82761 
9-82  775 
9.82  788 
9.82  802 
9.82816 

U 
14 
H 
14 
H 
13 
H 
H 

9-95  723 
9-95  748 
9-95  774 
9-95  799 
9-95  825 
9-95  850 

9-95  875 
9.95  901 
9-95  926 

25 
26 

25 
26 
25 
25 
26 
25 

0.04  277 
0.04  252 
0.04  226 

O.04  201 

0.04175 

0.04  150 
0.04  1  25 
0.04  099 
0.04  074 

9.86982 
9.86  970 
9.86  959 

9-86  947 
9.86  936 
9.86  924 
9.86913 
9.86  902 
9.86  890 

12 
II 
12 
II 
12 
II 
II 
12 

49 
48 

47 
46 
45 
44 
43 
42 
4i 

8 
9 

i 

2 

3 

20.8   20.0 

23.4  22.5 

14   13 
1.4   1.3 

2.8    2.6 

4-2   3-9 

2O 

9.82  830 

9-95  952 

0.04  048 

9.86879 

40 

4 

5.6   5.2 

21 
22 

23 
24 

3 

27 
28 

29 

9.82  844 
9.82  858 
9.82872 
9.82  885 
9.82  899 
9.82913 
9.82927 
9.82941 
9-82955 

14 
14 
13 
14 
14 
H 
14 
H 

9-95977 
9.96  002 
9.96028 
9.96053 
9.96078 
9.96  104 
9.96  129 
9-96  155 
9.96  1  80 

25 
26 

25 
25 
26 

25 
26 
25 

0.04  023 
0.03  998 
0.03  972 

0.03  947 
0.03  922 
0.03  896 
0.03  871 
0.03  845 
0.03  820 

9.86  867 
9.86855 
9.86  844 
9.86  832 
9.86  821 
9.86  809 
9.86  798 
9.86  786 
9-86  775 

12 
II 
12 
II 
12 
II 
12 
II 
12 

39 
38 
37 
36 
35 
34 
33 
32 
3i 

61 

7 
8 

9 

i 

8.4   7.8 
9.8   9.1 

I  1.  2   IO-4 

12.6  11.7 
12   11 

1.2    I.I 

3O 

9.82  968 

9.96  205 

26 

0-03  795 

9-86  763 

II 

30 

2 

2-4    2.2 

31 
32 
33 

34 

ii 
IS 

39 

9.82  982 
9.82  996 
9.83010 
9.83  023 
9-83  037 
9.83051 
9.83  065 
9.83  078 
9.83  092 

H 
M 
13 
H 
H 
H 
13 

14 

14 

9.96  231 
9.96  256 
9.96  281 
9.96  307 
9-96332 
9.96357 
9-96  383 
9.96408 
9-96433 

25 
25 
26 

25 
25 
26 

25 

3 

0.03  769 
0.03  744 
0.03719 
0.03  693 
0.03  668 
0.03  643 
0.03617 
0.03  592 
0.03  567 

9-86  752 
9.86  740 
9.86  728 
9.86717 
9.86  705 
9.86  694 
9.86682 
9.86  670 
9.86  659 

12 
12 
II 
12 
II 
12 
12 
II 
12 

29 
28 
27 
26 
25 
24 
23 

22 
21 

3 
4 

i 

9 

3-6   3-3 
4.8   4.4 

6-°   5-5 
7.2   6.6 
8.4   7-7 
9.6   8.8 
10.8   9.9 

4O 

9.83  106 

9-96  459 

0.03  541 

9.86  647 

20 

41 
42 

43 
44 

$ 
% 

49 

9.83  120 

9.83  133 
9.83  147 

9.83  161 
9-83  174 
9.83  1  88 

9.83  202 
9-832I5 
9.83  229 

13 
H 
H 
13 
H 
14 
13 
14 

9-96484 
9.96510 

9-96  535 
9-96  560 
9.96  586 
9.96611 
9.96  636 
9.96  662 
9.96  687 

26 
25 
25 
26 
25 
25 
26 
25 

0.03  5  1  6 
0.03  490 
0.03  465 
0.03  440 
0.03414 
0.03  389 
0.03  364 
0.03  338 
0.03313 

9.86635 
9.86  624 
9.86612 
9.86600 
9.86  589 
9.86577 
9-86  565 
9-86554 
9.86  542 

II 
12 
12 
II 
12 
12 
II 
12 

19 
18 
17 
16 
15 
H 
13 

12 
II 

o 
i 

2 

3 

* 

12  11   11 
26  26  25 

I.I   1.2   I.I 

3-2  3-5  3-4 
5-4  5-9  5-7 
7.6  8.3  8.0 

50 

9.83  242 

9.96712 

26 

0.03  288 

9.86  530 

IO 

|l 

1.9  13.0  12.5 

5i 
52 
53 
54 

I 

58 
59 

9.83  256 
9.83  270 
9.83  283 
9.83  297 
9-833'0 
9-83  324 
9-83  338 

9.8335I 
9-83  365 

H 
'3 
H 
13 
H 
14 
13 
H 

9.96  738 
9.96  763 
9.96  788 
9.96814 
9.96  839 
9.96  864 
9.96  890 
9.96915 
9.96  940 

25 
25 
26 

25 
25 
26 

25 
25 

0.03  262 
0.03  237 

0.03  212 

0.03  1  86 
0.03  161 
0.03  136 
0.03  1  10 
0.03  085 
0.03  060 

9.86518 
9-86  507 
9.86  495 
9.86  483 
9.86  472 
9.86  460 
9.86  448 
9-86436 
9-86425 

II 

12 
12 
II 
12 
12 
12 
II 

9 
8 
7 
6 
5 
4 
3 

2 

I 

°i 
c  I 

1 

^.i  15.4  14.8 
5.2  17.7  17.0 
B.4  20.1  19.3 
D.6  22.5  21.6 
2.8  24.8  23.9 
t-9  —  — 

6O 

9-83  378 

9.96  966 

0.03  034 

9.86413 

O 

L.  Cos. 

d. 

L.  Cot. 

c.  d. 

L.  Tan. 

L.  Sin. 

d. 

' 

P.P. 

47° 


43C 


4(J3 


1  r 

L.  Sin.  |  d. 

L.  Tan.  c.  d.  L.  Cot. 

L.  Cos.   d. 

p.  p. 

o 

I 

2 

3 

4 

I 

7 
8 
9 
IO 
ii 

12 
13 
H 

!i 

17 

19 

9-83  378 

14 
13 
H 
i3 
14 
13 
H 
13 
H 

9.96  966 

25 

11 

25 

11 

25 
25 

% 

25 

11 

25 
25 
26 

25 

11 

25 
25 
26 
25 

25 

3 

25 

11 

25 
25 
26 
25 
25 
25 
26 

25 
25 
26 
25 

25 

26 
25 
25 

•3 

25 

3 

25 
25 

25 
26 

25 

M 

25 
25 
25 

26 

0.03  034 
0.03  009 

O.O2  984 
0.02  958 

0.02  933 

0.02  908 
0.02  882 
0.02857 
O.O2  832 
O.O2  807 

9.86413 

12 
12 
12 

II 
12 
12 
12 
12 
12 
II 
12 
12 
12 
12 
12 
12 
12 

" 

12 
12 
12 
12 
12 
12 
12 
12 
12 
12 
12 
12 
12 
12 
12 
12 
12 
12 
12 
12 
12 
12 
12 
12 
12 
12 
12 
12 

13 
12 
12 
12 
12 
12 
12 

13 

12 
12 

12 

\2 
13 

60 

59 
58 
57 
56 
55 
54 
53 
52 
5i 
50 
49 
48 
47 
46 
45 
44 

43 

42 

4i 
40 

39 
38 
37 
36 
35 
34 
33 
32 
3i 
30 

29 
28 

27 
26 
25 
24 
23 

22 
21 

20 

19 
18 
17 
16 
15 
H 
U 

12 
II 

IO 

9 
8 
7 
6 
5 
4 
3 

2 

O 

2 

3 

4 

I 

I 

9 
i 

2 

3 
4 

i 

I 

9 
i 

2 

3 
4 

9 

26   25 

2.6   2.5 
S-2   5-° 
7-8   7-5 
10.4  i  o.o 

13.0  12.5 

15.6  15.0 
18.2  17.5 

20.8   2O.O 

23.4  22.5 

14   13 
1.4   1.3 

2.8    2.6 

4-2   3-9 
5-6   5-2 
7-o   6.5 
8.4   7.8 
9.8   9.1 

1  1.  2   IO-4 

12.6  11.7 
12  11 

1.2   I.I 

2.4   2.2 

3-6  3-3 
4.8  4.4 
6.0  5.5 
7.2  6.6 
8.4  7.7 
9.6  8.8 
10.8  9.9 

9-83  392 
9-83  405 
9.83419 

9-83  432 
9.83  446 
9-83  459 

9-83  473 
9.83  486 
9.83  500 

9-835U 
9-83  S2? 
9.83  540 
9-S3  554 

9-83  567 
9.83  581 

9-83  594 
9.83  608 
9.83  621 
9-83  634 

9.96991 
9.97016 
9-97  042 
9-97  °67 
9-97  092 
9.97118 

9-97  »43 
9.97  168 

9-97  193 

9.86  401 
9.86  389 
9.86377 
9.86  366 
9-86  354 
9-86  342 
9-86  330 
9.86318 
9.86  306 
9-86  295 

9.97  219 

O.O2  781 

13 
H 
13 
H 
J3 
14 
i3 
i3 
H 
13 
13 
14 
13 
H 
U 
13 
H 
13 
!3 
H 
13 
13 
13 
H 
13 
13 
i3 
H 
13 
13 
13 
14 
13 
13 
13 
13 
H 
13 
13 
13 
13 
13 
13 
H 
13 
»3 
'3 
i3 
U 

9-97  244 
9-97  269 
9.97  295 

9-97  320 
9-97  345 
9-97371 
9-97  396 
9.97421 

9-97  447 
9-97  472 
9-97  497 
9-97  523 
9-97  548 
9-97  573 
9-97  598 
9-97  624 
9-97  649 
9-97  674 
9.97  700 

O.O2  756 
0.02  731 
0.02  705 
0.02  680 
O.O2  655 
O.O2  629 
O.02  604 

0.02  579 
0.02  553 

9.86  283 
9.86271 
9-86  259 
9.86  247 
9.86235 
9.86  223 
9.86211 
9.86  200 
9.86  188 

20 

9.83  648 

O.O2  528 

9.86  176 

21 
22 
23 
24 

s 

27 
28 
29 

3O 

31 
32 
33 
34 

3 

37 
3* 
39 
40 

41 

42 

43 
44 

J2 

47 
48 

49 
50 

51 

52 
53 
54 

H 
H 

59 
60 

9.83  661 
9.83  674 
9.83  688 
9.83  701 

9-83715 
9.83  728 

9-83  741 
9-83  755 
9.83  768 

0.02  503 

0.02  477 
0.02452 

O.O2  427 
O.O2  4O2 
O.02  376 

0.02351 

O.02  326 
0.02  30O 

9.86  164 
9.86  152 
9.86  140 
9.86  128 
9.86  116 
9.86  104 
9.86  092 
9.86  080 
9.86  068 

9.83  781 

9-97  725 
9-97  75° 
9-97  776 
9.97  801 

9.97  826 
9.97851 
9-97  877 
9-97  902 
9-97  927 
9-97953 

O.O2  275 

9.86  056 

9-83  795 
9.83  808 
9.83  821 

9-83  834 
9.83  848 
9.83  861 
9.83  874 
9.83  887 
9-8390I 
9.83914 

O.02  250 
O.O2  224 
O.O2  199 
O.02  174 
O.O2  149 
O.02  123 
O.O2  098 
O.02  073 
0.02  047 

9.86  044 
9.86032 

9-86  020 
9-86  008 
9.85  996 
9.85  984 

9-85  972 
9.85  960 
9.85  948 

9-97  978 

O.02  022 

9.85  936 

13   13   12 
26  25   25 

1.0   I.O   I.O 

2   3-0  2.9  3.1 
5.0  4.8  5-2 
•»   7.0  6.7  7.3 
*   9-0  8.7  9.4 
?  i  i.o  10.6  n,j 
13.0  12.5  13.5 
7  15.0  14.4  15.6 
17.0  16.3  17.7 
jj  19.0  18.3  19.8 

2I.O  20.2  21.9 
23.0  22.1  24.0 
1*   25.0  24.0  — 

9.83927 
9.83  940 
9-83  954 
9.83  967 
9.83  980 
9-83  993 
9.84  006 
9.84  020 
9-84033 

9.98  003 
9.98029 
9.98054 
9.98079 
9.98  104 
9.98  130 

9-98  155 
9.98  1  80 
9.98  206 

o.o  i  997 
o.oi  971 
o.o  i  946 
o.oi  921 
o.oi  896 
o.oi  870 
o.oi  843 
o.oi  820 
o.oi  794 

9.85  924 
9.85912 
9.85  900 

9.85  888 
9.85  876 
9.85  864 
9.85851 
9.85  839 
9.85  827 

9.84  046 
9.84059 
9.84072 
9.84085 
9.84098. 

9.84  112 
9.84  125 
9.84  I38 
9.84  151 
9.84  164 

"9-84I77 

9-98231 

o.oi  769 

9-85815 

9.98  256 
9.98  281 
9.98  307 
9-98  332 
9-9-S  3S7 
9.98  383 
9.98  408 
9-98  433 
9-98  458 
9.98  484 

o.oi  744 
o.oi  719 
o.oi  693 
o.oi  668 
o.oi  643 
o.oi  617 
o.oi  592 
o.oi  567 
o.oi  ;  }_' 
o.oi  S"' 

9.85  803 
9-85  79i 
9-85  779 
9.85  766 
9-85  754 
9.85  742 

9-85  730 
9.85718 
9.85  706 

~9-85  693 

L.  Cos.   d. 

L.  Cot.  c.  d.|  L.  Tan. 

L.  Sin.   d. 

'     p.p. 

46° 


464 


44' 


L.  Sin. 

d. 

L.  Tan. 

c.  d. 

L.  Cot. 

L.  Cos. 

d. 

P.P. 

O 

9.84  177 

9.98  484 

o.oi  516 

9.85  693 

60 

I 

2 

3 

4 
5 
6 

7 
8 
9 

9.84  190 
9.84  203 
9.84  216 
9.84  229 
9.84  242 
9-^4  255 
9.84  269 
9.84  282 
9.84  295 

13 
13 
13 
13 
13 
H 
13 
i3 

9.98  509 
9-98  534 
9.98  560 

9.98  585 
9.98  610 
9.98  635 
9.98  661 
9.98  686 
9.98711 

3 

25 
25 
25 
26 

25 
25 

o.o  i  491 
o.oi  466 
o.oi  440 
o.oi  415 
o.oi  390 
o.oi  365 
o.oi  339 
o.oi  314 
o.oi  289 

9.85  68  1 
9.85  669 
9-85  657 

9-85  645 
9.85  632 
9.85  620 
9.85  608 
9.85  596 
9.85  583 

12 
12 
12 

13 
12 
12 
12 

59 
58 
57 
56 
55 
54 
53 
52 
5i 

I 

2 

3 

4  i 

I  ! 

7  i 

26  25   14 

2.6  2.5  1.4 
5.2  5.0  2.8 
7-8  7-5  4-2 
0.4  10.0  5.6 
3.0  12.5  7.0 
5.6  15.0  8.4 
8.2  17.5  9.8 

IO 

9.84  308 

9-98  737 

o.oi  263 

9.85  57i 

5O 

ii 

12 
13 
H 

!i 

3 

19 

9.84321 
9-84  334 
9-84  347 
9.84  360 
9-84  373 
9.84  385 

9.84  398 
9.84411 
9.84  424 

13 
i3 
13 
13 

12 
13 
13 
U 

9.98  762 
9.98  787 
9.98  812 
9.98  838 
9.98  863 
9.98  888 
9.98913 
9-98  939 
9.98  964 

25 
25 
26 

25 
25 
25 
26 
25 

o.oi  238 
o.oi  213 
o.oi  1  88 
o.oi  162 
o.oi  137 

O.OI  112 

o.oi  087 
o.oi  06  1 
o.oi  036 

9-85  559 
9-85  547 
9-85  534 
9.85  522 
9.85510 
9-85  497 
9-85  485 
9-85  473 
9.85  460 

12 
13 
12 
12 
13 
12 
12 
13 

49 
48 
47 
46 
45 
44 

43 

42 

4i 

I 

2 

3 
4 

i 

13   12 

1.3    1.2 

2.6   2.4 
3-9   3-6 
5.2   4.8 
6.5   6.0 
7.8   7.2 

20 

9-84  437 

9.98  989 

26 

O.OI  OI  I 

9.85  448 

4O 

7 

9.1   8.4 

21 
22 
23 

24 

9.84  450 
9.84  463 
9.84  476 

9-84489 

13 
13 
13 

n 

9.99015 
9.99  040 
9-99  065 
9.99  090 

25 
25 
25 
26 

o.oo  985 
o.oo  960 
o.oo  933 
0.00910 

9.85  436 
9.85  423 
9.85411 

9-85  399 

13 
12 
12 
13 

39 
38 
37 
36 

8 
9 

10.4   9.6 
11.7  10.8 

2§ 

26 

3 

29 

9.84  502 
9.84515 
9-84528 
9-84  54° 
9.84553 

13 

J3 

12 
13 

9-99  Hi 
9.99  1  66 
9.99  191 
9-99217 

25 
25 
25 
26 

o.oo  859 
o.oo  834 
o.oo  809 
o.oo  783 

9.55  350 
9-85  374 
9.85  361 
9-85  349 
9.85  337 

12 

13 
12 

12 

35 

34 
33 
32 
3i 

13   13 
26   25 

30 

9.84  566 

9.99  242 

o.oo  758 

9-85  324 

30 

o 

I  O    I.O 

3i 
32 
33 
34 
35 
36 

ii 

39 

9-84579 
9.84  592 
9.84  605 

9.84618 
9.84  630 
9.84  643 

9-84  656 
9.84  669 
9.84  682 

»3 
13 
13 
12 
13 
13 
13 
13 

9-99  267 
9.99  293 
9-99  3i8 
9-99  343 
9-99  368 
9-99  394 
9.99419 

9-99  444 
9.99  469 

26 

25 
25 
25 
26 

25 
25 

3 

o.oo  733 
o.oo  707 
o.oo  682 
0.00657 
o.oo  632 
o.oo  606 
o.oo  581 
o.oo  556 
o.oo  531 

9.85312 
9.85  299 
9.85  287 

9-85  274 
9.85  262 
9.85  250 
9.85  237 
9.85  225 

9.85  212 

13 

12 

13 

12 
12 

13 
12 
13 

29 
28 
27 
26 
25 
24 
23 

22 
21 

2 

3 

4 

7 
8 
9 

10 

3.0   2.9 
5.0   4.8 
7.0   6.7 
9.0   8.7 
i  i.o  10.6 
13.0  12.5 
15.0  14.4 
17.0  16.3 
19.0  18.3 

40 

9.84  694 

9-99  495 

o.oo  505 

9.85  2OO 

2O 

ii 

4i 
42 

43 
44 
45 
46 

47 
48 
49 

9.84  707 
9.84  720 
9-84  733 
9.84  745 
9.84  758 
9.84771 
9.84  784 
9.84  796 
9.84809 

13 
13 
12 

13 
13 
13 
12 
13 

I  7 

9-99  520 
9-99  545 
9-99  57° 
9.99  596 
9.99  621 
9.99  646 

9-99  672 
9.99  697 
9-99  722 

25 
25 
26 

25 
25 
26 

25 
25 

o.oo  480 
o.oo  453 
o.oo  430 
o.oo  404 
o.oo  379 
o.oo  354 
o.oo  328 
o.oo  303 
o.oo  278 

9.85  I87 

9.85175 
9.85  l62 

9.85  I^O 
9.85  137 
9.85  123 
9.85  112 

9.85  too 

9.85  087 

12 
13 
12 

13 
12 

13 
12 
13 

19 

18 
J7 
16 
15 
H 
13 

12 

n 

12 
»3 

O 
I 

2 

25.0  24.0 

12   12 
26   25 

I.I    I.O 

3-2   3-i 

5.4   5.2 

5O 

9.84  822 

1  -1 

9-99  747 

"A 

o.oo  253 

9.85  074 

10 

3 

7-6   7-3 

5i 

52 
53 
54 
55 
56 

P 

59 

9.84  835 
9.84  847 
9.84  860 
9.84  873 
9.84  885 
9.84  898 
9.84911 
9.84  923 
9.84  936 

12 
»3 
13 
12 
13 
13 
12 
13 

I? 

9-99  773 
9-99  798 
9-99  823 
9-99  848 
9-99  874 
9.99  899 

9-99  924 
9.99  949 

9-99  975 

25 
25 
25 
26 
25 
25 

3 

o.oo  227 

O.OO  2O2 

o.oo  177 
o.oo  152 
o.oo  126 

O.OO  101 

0.00076 
o.oo  05  1 
o.oo  025 

9.85  062 
9.85  049 
9.85  037 
9.85  024 
9.85012 

9-84  999 
9.84  986 
9.84  974 
9.84961 

13 
12 

13 
12 
13 
13 
12 
13 

9 

8 

7 
6 
5 
4 
3 

2 

I 

5 
6 

I 

9 

10 

ii 

12 

9.8   9.4 
11.9  11.5 
14.1  13.5 
16.2  15.6 
18.4  17-7 
20.6  19.8 

22.8   21-9 
24.9   24.0 

60 

9.84  949 

o.oo  ooo 

0.00  OOO 

9.84  949 

O 

L.  Cos. 

d. 

L.  Cot. 

c.  d. 

L.  Tan. 

L.  Sin. 

d. 

' 

P.P. 

45' 


TABLES    XVII.,    XVIII. 

NATURAL   TRIGONOMETRIC    FUNCTIONS. 


K'M'H  st'Rv.  — 30  465 


466 


TABLE   XVII.  —  NATURAL  SIXKS  AND  COSINKS. 
Natural  Sines. 


Prop. 

Angle. 

0' 

10' 

20' 

30' 

40' 

50' 

60' 

Angle. 

Parts 

forV. 

0° 

.000000 

.002909 

.0058  1  8 

.0087  27 

.011635 

.0145  44 

.017452 

89° 

2-9 

i 

.017452 

.0203  6 

.0232  7 

.0261  8 

.0290  8 

.03199 

.0349  o 

88 

2.9 

2 

.0349  o 

.0378  i 

.0407  i 

.0436  2 

.0465  3 

.0494  3 

•05234 

87 

2.9 

3 

•0523  4 

•05524 

.0581  4 

.0610  5 

.0639  5 

.0668  5 

.0697  6 

86 

2.9 

4 

.0697  6 

.0726  6 

•0755  6 

.0784  6 

.0813  6 

.0842  6 

.0871  6 

85 

2.9 

5 

.0871  6 

.09005 

.09295 

•09585 

.0987  4 

.10164 

•1045  3 

84 

2.9 

6 

•10453 

.10742 

.1103  i 

.11320 

.11609 

.11898 

.12187 

83 

2.9 

7 

.12187 

.12476 

.12764 

•1305  3 

•1334 

•1363 

.1392 

82 

2.9 

8 

.1392 

.1421 

.1449 

.1478 

•1507 

•1536 

.1564 

81 

2.9 

9 

.1564 

•1593 

.1622 

.1650 

.1679 

.1708 

•1736 

80 

2-9 

IO 

.1736 

.1765 

•1794 

.1822 

.1851 

.1880 

.1908 

79 

2-9 

ii 

.1908 

•1937 

.1965 

.1994 

.2022 

.2051 

.2079 

78 

2.9 

12 

.2079 

.2108 

.2136 

.2164 

•2193 

.2221 

.2250 

77 

2.8 

13 

.2250 

.2278 

.2306 

•2334 

•2363 

.2391 

.2419 

76 

2.8 

H 

.2419 

.2447 

.2476 

.2504 

•2532 

.2560 

.2588 

75 

2.8 

15 

.2588 

.2616 

.2644 

.2672 

.2700 

.2728 

.2756 

74 

2.8 

16 

.2756 

.2784 

.2812 

.2840 

.2868 

.2896 

.2924 

73 

2.8 

X7 

.2924 

.2952 

•2979 

.3007 

.3035 

.3062 

.3090 

72 

2.8 

18 

.3090 

.3118 

•3H5 

•3J73 

•3201 

.3228 

•3256 

7i 

2.8 

19 

.3256 

•3283 

•33" 

.3338 

•3365 

•3393 

.3420 

70 

2.7 

20 

•3420 

•3448 

•3475 

.3502 

•3529 

•3557 

•3584 

69 

2.7 

21 

.3584 

.3611 

•3638 

•3665 

.3692 

•3719 

•3746 

68 

2-7 

22 

•3746 

•3773 

.3800 

.3827 

•3854 

•3881 

•3907 

67 

2.7 

23 

•3907 

•3934 

.3961 

•3987 

.4014 

.4041 

.4067 

66 

2-7 

24 

.4067 

.4094 

.4120 

.4147 

•4173 

.4200 

.4226 

65 

2-7 

25 

.4226 

•4253 

•4279 

•4305 

•4331 

•4358 

4384 

64 

2.6 

26 

.4384 

.4410 

•4436 

.4462 

.4488 

•45  H 

•454° 

63 

2.6 

27 

•4540 

.4566 

•4592 

.4617 

•4643 

.4669 

•4695 

62 

2.6 

28 

•4695 

.4720 

.4746 

.4772 

•4797 

.4823 

.4848 

61 

2.6 

29 

.4848 

.4874 

.4899 

.4924 

•495° 

•4975 

.5000 

60 

2-5 

3O 

.5000 

•5025 

•5°5° 

•5075 

.5100 

•5125 

•5150 

59 

2-5 

3i 

•S^o 

•S17S 

.5200 

•5225 

•5250 

•5275 

•5299 

58 

2-5 

32 

•5299 

•5324 

•5348 

•5373 

•5398 

.5422 

•5446 

57 

2-5 

33 

.5446 

•5471 

•5495 

•55i9 

•5544 

.5568 

•5592 

56 

2.4 

34 

•5592 

.5616 

.5640 

.5664 

.5688 

•5712 

•5736 

55 

2.4 

35 

•5736 

.5760 

.5783 

.5807 

•5831 

•5854 

.5878 

54 

2.4 

36 

.5878 

.5901 

•5925 

.5948 

•5972 

•5995 

.6018 

53 

2-3 

37 

.6018 

.6041 

.6065 

.6088 

.6m 

.6134 

•6i57 

52 

2-3 

38 

•6i57 

.6180 

.6202 

.6225 

.6248 

.6271 

.6293 

51 

2-3 

39 

.6293 

.6316 

•6338 

.6361 

•6383 

.6406 

.6428 

50 

2.3 

4O 

.6428 

.6450 

.6472 

.6494 

.6517 

•6539 

.6561 

49 

2.2 

4i 

.6561 

.6583 

.6604 

.6626 

.6648 

.6670 

.6691 

48 

2.2 

42 

.6691 

•6713 

•6734 

.6756 

.6777 

.6799 

.6820 

47 

2.2 

43 

.6820 

.6841 

.6862 

.6884 

.6905 

.6926 

.6947 

46 

2.1 

44 

.6947 

.6967 

.6988 

.7009 

.7030 

.7050 

.7071 

45 

2.1 

60' 

50' 

40' 

30' 

20' 

10' 

0' 

Angle. 

Natural  Cosines. 


KAI,  SINKS  AM>  COSINKS  (conti 
Natural   Sines. 


467 


Prop. 

Angle 

0' 

10' 

20' 

30' 

40' 

50' 

60' 

Angle. 

Parts 

forV. 

45° 

.7071 

.7092 

.7112 

•7133 

.7153 

•7173 

•7'93 

44° 

2.O 

46 

•7'93 

.7214 

•7234 

•7254 

.7274 

•7294 

•73H 

43 

2.O 

47 

•73M 

•7333 

•7353 

•7373 

•7392 

.7412 

•7431 

42 

2.O 

48 

•7431 

•7451 

.7470 

.7490 

.7509 

•7528 

•7547 

4i 

1-9 

49 

•7547 

.7566 

•7585 

.7604 

.7623 

.7642 

.7660 

40 

1-9 

5O 

.7660 

.7679 

.7698 

.7716 

•7735 

•7753 

.7771 

39 

•9 

51 

.7771 

.7790 

.7808 

.7826 

.7844 

.7862 

.7880 

38 

.8 

52 

.7880 

.7898 

.7916 

•7934 

•795i 

.7969 

.7986 

37 

.8 

53 

.7986 

.8004 

.8021 

.8039 

.8056 

•8073 

.8090 

36 

•7 

54 

.8090 

.8107 

.8124 

.8141 

.8158 

•8i75 

.8192 

35 

•7 

55 

.8192 

.8208 

.8225 

.8241 

.8258 

.8274 

.8290 

34 

1.6 

56 

.8290 

•8307 

•8323 

•8339 

•8355 

•8371 

•8387 

33 

1.6 

57 

.8387 

.8403 

.8418 

•8434 

.8450 

.8465 

.8480 

32 

1.6 

58 

.8480 

.8496 

.8511 

.8526 

.8542 

•8557 

•8572 

3i 

i-5 

59 

•8572 

.8587 

.8601 

.8616 

•8631 

.8646 

.8660 

30 

1-5 

60 

.8660 

•8675 

.8689 

.8704 

.8718 

•8732 

.8746 

29 

1.4 

61 

.8746 

.8760 

.8774 

.8788 

.8802 

.8816 

.8829 

28 

1.4 

62 

.8829 

•8843 

•8857 

.8870 

.8884 

.8897 

.8910 

27 

1.4 

63 

.8910 

.8923 

.8936 

•8949 

.8962 

•8975 

.8988 

26 

1-3 

64 

.8988 

.9001 

.9013 

.9026 

.9038 

.9051 

.9063 

25 

i-3 

65 

.9063 

•9075 

.9088 

.9100 

.9112 

.9124 

•9135 

24 

1.2 

66 

•9  '35 

.9147 

•9i59 

.9171 

.9182 

.9194 

.9205 

23 

1.2 

67 

.9205 

.9216 

.9228 

•9239 

.9250 

.9261 

.9272 

22 

I.I 

68 

.9272 

.9283 

•9293 

•9304 

•9315 

•9325 

•9336 

21 

I.I 

69 

•9336 

•9346 

•9356 

•9367 

•9377 

.9387 

•9397 

20 

I.O 

7O 

•9397 

.9407 

.9417 

.9426 

•9436 

.9446 

•9455 

19 

I.O 

7' 

•9455 

•9465 

•9474 

•9483 

•9492 

.9502 

•95" 

18 

0.9 

72 

•95" 

.9520 

•9528 

•9537 

•9546 

•9555 

•9563 

17 

0.9 

73 

•9563 

•9572 

.9580 

.9588 

•9596 

.9605 

.9613 

16 

0.8 

74 

.9613 

.9621 

.9628 

•9636 

.9644 

.9652 

.9659 

15 

0.8 

75 

•9659 

.9667 

.9674 

.9681 

.9689 

.9696 

•97°3 

14 

0.7 

76 

.9703 

.9710 

.9717 

.9724 

•973° 

•9737 

•9744 

13 

0.7 

77 

•9744 

•9750 

•9757 

•9763 

.9769 

•9775 

.9781 

12 

0.6 

78 

.9781 

.9787 

•9793 

•9799 

.9805 

.9811 

.9816 

II 

0.6 

79 

.9816 

.9822 

.9827 

•9833 

.9838 

•9843 

.9848 

IO 

o-5 

80 

.9848 

•9853 

.9858 

•9863 

.9868 

.9872 

.9877 

9 

o-5 

81 

•9877 

.9881 

.9886 

.9890 

.9894 

.9899 

•9903 

8 

0.4 

82 

•9903 

.9907 

.9911 

.9914 

.9918 

.9922 

•9925 

7 

0.4 

83 

•9925 

.9929 

•9932 

•9936 

•9939 

•9942 

•9945 

6 

o-3 

84 

•9945 

.9948 

•995  * 

•9954 

•9957 

•9959 

.9962 

'  5 

0-3 

85 

.9962 

.9964 

.9967 

.9969 

•997  1 

•9974 

.9976 

4 

O.2 

86 

.9976 

.9978 

.9980 

.9981 

.9983 

.9985 

.9986 

3 

0.2 

87 

.9986 

.9988 

•9989 

.9990 

.9992 

•9993 

•9994 

2 

O.I 

88 

•9994 

•9995 

.9996 

•9997 

•9997 

•9998 

•9998 

I 

O.J 

89 

.9998 

•9999 

•9999 

I.OOOO 

I.OOOO 

I.OOOO 

I.OOOO 

O 

o.o 

60' 

50' 

40' 

30' 

20' 

10' 

0' 

Angle. 

Natural  Cosines. 


468  TABLE   XVIII.  —  XATI-KAI.  TAN<;I:MS  AXD  COTAXCJKNTS. 

Natural  Tangents. 


Prop. 

Angle 

0' 

10' 

20' 

30' 

40' 

50' 

60' 

Angle 

Parts 

for  1'. 

0° 

.00000 

.0029  i 

.0058  2 

.00873 

.01164- 

•0145  5 

.01746 

89° 

2-9 

i 

.01746 

.0203  6 

.0232  8 

.0261  9 

.0291  o 

.0320  i 

.0349  2 

88 

2.9 

2 

.0349  2 

•0378  3 

.0407  5 

.0436  6 

.0465  8 

.0494  9 

.0524  I 

87 

2.9 

3 

.0524  I 

•0553  3 

.0582  4 

.061  1  6 

.0640  8 

.0670  o 

.0699  3 

86 

2-9 

4 

.0699  3 

.0728  5 

•0757  8 

.0787  o 

.08163 

.0845  6 

.0874  9 

85 

2.9 

5 

.0874  9 

.0904  2 

•0933  5 

.0962  9 

.09923 

.1021  6 

.1051  o 

84 

2.9 

6 

.1051  o 

.10805 

.11099 

•  "394 

.11688 

.11983 

.12278 

83 

2.9 

7 

.12278 

•12574 

.12869 

•13165 

.1346 

•1376 

.1405 

82 

3-o 

8 

.1405 

•H35 

.1465 

•H95 

•1524 

•1554 

.1584 

81 

3-o 

9 

.1584 

.1614 

.1644 

•1673 

•1703 

•1733 

•'763 

80 

3-o 

10 

•!763 

•1793 

.1823 

•1853 

.1883 

.1914 

.1944 

79 

3-o 

ii 

.1944 

.1974 

.2004 

.2035 

•2065 

.2095 

.2126 

78 

3-o 

12 

.2126 

.2156 

.2186 

.2217 

•2247 

.2278 

.2309 

77 

3-i 

'3 

.2309 

•2339 

.2370 

.2401 

.2432 

.2462 

•2493 

76 

3-i 

H 

•2493 

•2524 

•2555 

.2586 

.2617 

.2648 

.2679 

75 

3-i 

15 

.2679 

.2711 

.2742 

•2773 

.2805 

.2836 

.2867 

74 

3-i 

1  6 

.2867 

.2899 

.2931 

-.2962 

.2994 

.3026 

•3057 

73 

3-2 

i? 

•3057 

.3089 

.3121 

•3153 

•3185 

•3217 

•3249 

72 

3-2 

18 

•3249 

.3281 

•3314 

•3346 

•3378 

•34" 

•3443 

7* 

3-2 

19 

•3443 

•3476 

.3508 

•3541 

•3574 

•3607 

.3640 

70 

3-3 

20 

.3640 

•3673 

.3706 

•3739 

•3772 

.3805 

•3839 

69 

3-3 

21 

•3839 

.3872 

.3906 

•3939 

•3973 

.4006 

.4040 

68 

3-4 

22 

.4040 

.4074 

.4108 

.4142 

.4176 

.4210 

•4245 

67 

3-4 

23 

•4245 

•4279 

•43H 

•4348 

•4383 

.4417 

•4452 

66 

3-5 

24 

.4452 

.4487 

.4522 

•4557 

•4592 

.4628 

•4663 

65 

3-5 

25 

.4663 

.4699 

•4734 

•477° 

.4806 

.4841 

.4877 

64 

3-6 

26 

•4877 

.4913 

•495° 

.4986 

.5022 

•5059 

•5°95 

63 

3-6 

27 

•5°95 

•5132 

.5169 

•5206 

•5243 

.5280 

•5317 

62 

3-7 

28 

•5317 

•5354 

•5392 

•543° 

•5467 

•55°5 

•5543 

61 

3-8 

29 

•5543 

•558i 

.5619 

.5658 

.5696 

•5735 

•5774 

60 

3-8 

30 

•5774 

.5812 

•5851 

.5890 

•5930 

•5969 

.6009 

59 

3-9 

3i 

.6009 

.6048 

.6088 

.6128 

.6168 

.6208 

.6249 

58 

4.0 

32 

.6249 

.6289 

•633° 

•6371 

.6412 

•6453 

•6494 

57 

4.1 

33 

.6494 

.6536 

•6577 

.6619 

.6661 

.6703 

•6745 

56 

4.2 

34 

•6745 

.6787 

.6830 

•6.873 

.6916 

•6959 

.7002 

55 

4-3 

33 

.7002 

.7046 

.7089 

•7*33 

.7177 

.7221 

.7265 

54 

4-4 

36 

.7265 

•7310 

•7355 

.7400 

•7445 

.7490 

•7536 

53 

4-5 

37 

•7536 

•758i 

.7627 

•7673 

.7720 

.7766 

•7813 

52 

4.6 

38 

•78i3 

.7860 

.7907 

•7954 

.8002 

.8050 

.8098 

5i 

4-7 

39 

.8098 

.8146 

•8i95 

•8243 

.8292 

.8342 

•8391 

5° 

4-9 

4O 

•8391 

.8441 

.8491 

.8541 

•8591 

.8642 

•8693 

49 

5-° 

4i 

.8693 

.8744 

.8796 

.8847 

.8899 

•8952 

.9004 

48 

5-2 

42 

.9004 

.9057 

.9110 

•9163 

.9217 

.9271 

•9325 

47 

5-4 

43 

•9325 

.9380 

•9435 

.9490 

•9545 

.9601 

•9657 

46 

5'5 

44 

.9657 

•9713 

.9770 

•9827 

.9884 

•9942 

I.OOOO 

45 

5-7 

60' 

50' 

40' 

30' 

20' 

10' 

0' 

Angle. 

Natural  Cotangents. 


NATCKAI.  TANUKMS  AND  COTAXGKNTS  (continued). 
Natural  Tangents. 


409 


Prop. 

Angle. 

0' 

10' 

20' 

30' 

40' 

50' 

60' 

Angle. 

Parts 

for  1'. 

45° 

I.OOOO 

1.0058 

1.0117 

1.0176 

1.0235 

1.0295 

'•0355 

44° 

5-9 

46 

r-°355 

1.0416 

1.0477 

1.0538 

1-0599 

1.  0661 

1.0724 

43 

6.1 

47 

1.0724 

1.0786 

1.0850 

1.0913 

1.0977 

.1041 

1.1106 

42 

6.4 

48 

1.1106 

1.1171 

1.1237 

1.1303 

1.1369 

.1436 

1.1504 

41 

6.6 

49 

1.1504 

I-I57I 

1.1640 

1.1708 

1.1778 

.1847 

1.1918 

40 

6-9 

5O 

1.1918 

1.1988 

1.2059 

1.2131 

1.2203 

.2276 

1-2349 

39 

7.2 

5* 

1.2349 

1.2423 

1.2497 

1.2572 

1.2647 

•2723 

1.2799 

38 

7-5 

52 

1.2799 

1.2876 

1.2954 

1.3032 

1.3111 

1.3190 

1.3270 

37 

7-9 

53 

1.3270 

I-3351 

1-3432 

L35H 

'•3597 

1.3680 

1.3764 

36 

8.2 

54 

I-3764 

1-3848 

1-3934 

1.4019 

1.4106 

I-4I93 

1.4281 

35 

8.6 

55 

1.4281 

1-437° 

1.4460 

1-4550 

1.4641 

1-4733 

1.4826 

34 

9-i 

56 

1.4826 

1.4919 

L50I3 

1.5108 

1.5204 

1.5301 

1-5399 

33 

9-6 

57 

1-5399 

1-5497 

1-5597 

1.5697 

I.5798 

1.5900 

1.6003 

32 

10.  1 

58 

1.6003 

1.6107 

1.6212 

1.6319 

1.6426 

1-6534 

1-6643 

3i 

10.7 

59 

1.6643 

I-6753 

1.6864 

1.6977 

1.7090 

1.7205 

1.7321 

30 

"•3 

6O 

1.7321 

L7437 

I.7556 

I-7675 

1.7796 

1.7917 

1.8040 

29 

I2.O 

61 

i  .8040 

1.8165 

1.8291 

1.8418 

1.8546 

1.8676 

1.8807 

28 

12.8 

62 

1.8807 

1.8940 

1.9074 

1.9210 

1-9347 

1.9486 

1.9626 

27 

I3.6 

63 

1.9626 

1.9768 

1.9912 

2.0057 

2.0204 

2-0353 

2.0503 

26 

14.6 

64 

2.0503 

2.0655 

2.0809 

2.0965 

2.1123 

2.1283 

2.1445 

25 

15-7 

65 

2.1445 

2.1609 

2-1775 

2.1943 

2.2113 

2.2286 

2.2460 

24 

16.9 

66 

2.2460 

2.2637 

2.2817 

2.2998 

2.3183 

2-3369 

2-3559 

23 

18.3 

67 

2-3559 

2.3750 

2-3945 

2.4142 

2-4342 

2-4545 

2-475  i 

22 

19.9 

68 

2.4751 

2.4960 

2.5172 

2.5386 

2-5605 

2.5826 

2.6051 

21 

21.7 

69 

2.6051 

2.6279 

2.6511 

2.6746 

2.6985 

2.7228 

2-7475 

2O 

23-7 

70 

2-7475 

2-7725 

2.7980 

2.8239 

2.8502 

2.8770 

2.9042 

19 

7i 

2.9042 

2.9319 

2.9600 

2.9887 

3-0178 

3-0475 

3.0777 

18 

72 

3-0777 

3.1084 

3-1397 

3.1716 

3.2041 

3-2371 

3.2709 

J7 

73 

3.2709 

3-3052 

3-3402 

3-3759 

3-4124 

3-4495 

34874 

16 

74 

3-4874 

3.5261 

3-5656 

3-6059 

3.6470 

3-6891 

3-7321 

'5 

75 

3-7321 

3.7760 

3.8208 

3-8667 

3-9I36 

3-9617 

4.0108 

14 

76 

4.0108 

4.0611 

4.1126 

4-1653 

4-2193 

4.2747 

4-3315 

»3 

77 

4.3315 

4-3897 

4-4494 

4-5107 

4.5736 

4.6382 

4.7046 

12 

78 

4.7046 

4-7729 

4.8430 

4.9152 

4.9894 

5.0658 

5.1446 

II 

79 

5.1446 

5-2257 

5-3093 

5-3955 

54845 

5-5764 

5-6713 

IO 

8O 

5-67I3 

5-7694 

5.8708 

5-9758 

6.0844 

6.1970 

6.3138 

9 

Si 

6.3138 

6.4348 

6.5606 

6.6912 

6.8269 

6.9682 

7.1154 

8 

82 

7-"54 

7.2687 

74287 

7-5958 

77704 

7-9530 

8.1443 

7 

83 

8.1443 

8.3450 

8-5555 

8.7769 

9.0098 

9-2553 

9-5*44 

6 

84 

9-5'44 

9.7882 

10.0780 

10.3854 

10.7119 

11.0594 

11.4301 

5 

85 

11.4301 

11.8262 

12.2505 

12.7062 

13.1969 

13.7267 

14.3007 

4 

86 

14.3007 

14.9244 

15.6048 

16.3499 

17.1693 

18.0750 

19.0811 

3 

87 

19.0811 

20.2056 

21.4704 

22.9038 

24.5418 

26.4316 

28.6363 

2 

88 

28.6363 

31.2416 

34-3678 

38.1885 

42.9641 

49.1039 

57.2900 

I 

89 

57.2900 

68.7501 

85.9398 

114.5887 

171.8854 

343-7737 

00 

O 

60' 

50' 

40' 

30' 

20' 

10' 

0' 

Angle. 

Natural  Cotangents. 


PLATE    I 


PLATE    I 


Itutial  Meridian  -  Madrid 


PLATE    III. 


, 

-  V 


VERTICAL  SECTION  EAST  AND  WEST. 


PLATE    IV. 


VERTICAL    SECTION 
NORTH  AND  SOUTH. 


Plan  and  Vertical  Sections 

of  the 

UNDERGROUND    WORKS 

CORA  BLANCA  MINE 

NEW  ALMADEN 

California 


Scale  in  feet 

100        ISO       100 


April,  1896 


Tlu  Pifura  in  braetttt,  tkut  (610),  inditat,  t*«  wrticat  dinne 
4./o«.  th,  datum  poi«t,  a  m,num.,,l  o*  1*.  ,.mmit  of  Mi,,  Hill 
flu  tkaft*  <txi  trvtlt  <m  (Ac  Stctims  art  prajKM  upon  ,„, 
ntatiu,  ilnviint. 


PLATE  V. 


PLATE  VI 


INDEX. 


Abney  level  and  clinometer,  72,  73. 
Abscissas,  defined,  156. 

Diagram  of  positive  and  negative, 

162. 

Accuracy  of  stadia  measurements,  127, 
139,  140. 

of  barometric  leveling,  76. 
Additions,  Planning  of,  235. 

Survey  and  map  of,  236. 

Plots  of,  354. 
Adit,  defined,  308. 

Mine  entered  by,  316. 
Adjustments,  of  level,  50,  63. 

Collimation  adjustment,  64. 

of  objective  slide,  65. 

of  bubble,  67. 

Peg  method,  68. 

Lateral  adjustment  of  bubble,  69. 

Y  adjustment,  70. 

of  dumpy  level,  70. 

of  any  level  on  metal  base,  71. 

of  Locke  and  Abney  levels,  73. 

of  compass,  79. 

of  transit,  108. 

of  solar  transit,  122. 

of  Saegmuller  solar  attachment,  125. 

of  triangles,  259. 

of  plane  table,  274. 

of  sextant,  297. 
Agonic  line,  87. 
Alidade,  of  transit,  96,  102,  126. 

of  Colby  protractor,  2<iti. 

of  plane  table,  268. 
Alignment  of  chain,  27. 
A  Urn's  stadia  rod,  131. 
Alluvion,  Apportionment  of,  348. 
Altitude,  Measuring  difference  of.    See 
"Leveling." 

Relation  to  barometer  readings,  75. 


Altitude  of  heavenly  bodies,  117,  125, 

364. 
Aneroid    barometer,    Leveling    with, 

74-76. 
Angles,  Measurement  of,  77-126,  254, 

257.    See  also  "  Direction." 
Precision  in  laying  off  of,  230. 
measured  by  repetition,  255,  256. 
measured  by   continuous  reading, 

255,  257. 

Angular  convergences, 
of  meridians,  224. 
and  distances  between  meridians. 

Table  of,  371. 

Annual  variation  of  declination,  86. 
Appendix,  322-469. 

Problems  and  examples,  322-340. 
Judicial  functions    of    surveyr-s, 

341-350. 

Ownership  of  surveys,  351-356. 
Geographical  positions  of  base  lines 
and  meridians  in  public  surveys, 
357-360. 
Tables,  361-469. 

Approximate  local  mean  times  of  cul- 
mination and  elongation  of  Po- 
laris, 365. 

Approximations  in  earthwork  compu- 
tations, 277. 

Arcs,  of  tachy meters,  99. 
located  by  chain,  242. 
and  chords,  243. 
Areas,  in  land  survey  computations, 

152-156. 

by  double  longitudes,  152. 
To  find  the  area  of  a  field,  154. 
Irregular  areas  by  offsets,  155. 
of  closed  surveys,  155,  160. 
by  coordinates,  159,  170. 


471 


472 


INDEX. 


Areas,  elongated,  by  offsets,  161. 

example  solved  by  latitude  differ- 
ences and  double  longitades, 
168. 

of  parcels  of  ground,  204. 

of  fields,  204,  205. 

Topographical  surveys  for  small, 
247. 

Topographical  surveys  for  large, 
248. 

to  be  triangulated,  253. 

of  similar  figures,  277. 

Grading,  279. 

Average  end,  279. 

of  cut  and  fill,  286. 
Atmosphere,  Pressure  of,  74. 
Attraction,  local,  93 . 
Average  end  areas,  279. 
Axis,  of  revolution,  Adjustment  of,  11  . 

Polar,  117,  119,  121. 

Polar,  Adjustment  of,  123. 
Axes  of  earth,  Lengths  of,  9. 
Azimuth,  at  elongation,  defined,  91. 

of  a  line,  defined,  102. 

Determination  of,  105,  260. 

Needle  checks  on,  107. 

Zero,  162. 

in  application  of  coordinates,  214. 

in  transit  and  stadia  method  of 
topography,  249. 

Plane  table  turned  in,  269. 

Back  bearing,  83,  84. 
Backsights,  51,  57. 

Baker's    "Engineer's    Surveying  In- 
struments," 46,  136. 
Balancing  surveys,  141,  144-148,  166, 
214. 

Fundamental  hypotheses,  144. 

Method  explained,  145. 

The  practice,  147. 

When  a  transit  is  used,  148. 

Example  in,  166. 
Barometer,  Leveling  with,  74-76. 

described,  74. 

Theory  of,  74. 

Barometric  tables,  75,  362,  363. 

Practical  suggestions,  76. 

Accuracy  of  method,  76. 
Base  lines,  in  U.  S.  public  land  sur- 
veys, 220,  223,  357-360. 

measured  in  triangulation,  253, 254. 

in  mapping,  261. 


Bay,  Dredging  of,  275. 
Bayous,  Meandering  of,  228. 
Beacons,  294. 
Bearings,  of  lines,  83. 

Errors  in,  145. 

Magnetic,  212. 

Bed  of  stream,  Ownership  of,  348. 
Bench  marks,  in  leveling,  54,  55. 

in  mines,  313. 
Binocular  hand  level,  73. 
Black  Hills  meridian,  360. 
Boise"  meridian,  359. 
Boston  leveling  rod,  47,  49. 
Boundaries,  of  land,  341,  343,  345, 351. 

of  water,  216,  217,  346,  347,  349. 
Bremicker  logarithmic  tables,  165. 
Brough's  ' '  Treatise  on  Mine  Survey- 
ing," 313. 
Bubble,  Level,  35-38. 

Adjustment  of;  64,  67. 

of  compass,  79. 

in  transit,  108. 

for  leveling  plane  table,  274. 
"  Bulletin  of  University  of  Wisconsin," 

139. 

Bull's-eye  lamp  for  transit,  106. 
Buoys    and    buoy    flags,    291,     292, 

294. 
Burt's  solar  compass,  116. 

Canals,  Survey  of,  238. 
Carpenter's  level,  71. 
Carpenter's  rule,  181. 
Cautions,  in  chaining,  21. 

in  use  of  aneroid  barometer,  76. 

in  use  of  compass,  85. 

in  use  of  solar  transit,  120. 
Center  of  gravity  of  prism  bases,  275. 
Chaining,  Method  of,  18-21. 

Hints  on,  21. 

on  slopes,  21,  27. 

Errors  in,  22. 

Chains,  Gunter's  and  engineer's,  13, 
14. 

Temperature  of,  24, 

Sag  and  pull  of,  25,  26. 

Alignment  of,  27. 

Gunter's,     reduced     to    feet    and 

meters,  377,  378. 
Chain  surveying,  204-208. 

Preliminary  examination,  205. 

Survey,  206. 

Notes,  208. 


INDEX. 


473 


Channels,  Sounding  in,  292. 

Check  mark  in  triangulation  notes,  256. 

Chickasaw  meridian,  359. 

Choctaw  meridian,  358. 

Cimarron  meridian,  360. 

Circle,  shape  of  latitude  parallels,  10. 

Circular  arcs  and  measures,  Tables  of, 

376. 
Circumference,  Zero,  178. 

of  planimeter  wheel,  178. 
City  lot,  Description  of,  233. 
City  surveying,  230-237. 

Precision  required,  230. 

Extent  of  survey,  231. 

Instruments,  231. 

Description  of  city  lot,  233. 

Finding  city  lot,  233. 

Marking  corners,  234. 

Discrepancies,  234. 

Planning  additions,  235. 

Making  survey  and  map  for  an  ad- 
dition, 236. 

Claims,  mining,  306,  307. 
Clarke's  spheroid,  of  1866,  9. 

of  1880,  9. 
Clinometer,  with  Abney  level,  72. 

Hanging,  313. 
Closed  fields,  Survey  of,  141,  155,  159, 

160,  205. 

Closing  line,  146,  263. 
Closure,  Error  of,  141,  143,  144,  145, 

171. 
Coal  mines,  Survey  of,  305. 

Mapping  of,  320. 
Coefficient,  correction,  361. 
Coefficient  of  expansion  of  steel,  24. 
Colatitude  reading,  121. 
Colby's  protractor,  265. 
Colby's  slide  rule,  136,  253. 
Collimation  adjustment  of  level,  64. 
Collimation  line,  defined,  45. 

Adjustment  of,  64,  113. 

Revolution  of,  109-112. 

in  solar  transit,  122. 
Common  logarithms  of  numbers,  Table 

of,  397-418. 

Compass,   77-85,    94,    104,    116,   205, 
213. 

Description,  77. 

Solar,  77,  116. 

Requirements  for  adjustments,  79. 

Plate  bubbles,  79. 

Sights,  80. 


Compass,  Needle  and  pivot,  80. 

Bearing,  83. 

To  determine  bearing  of  a  line,  83. 

To  lay  out  a  line  of  given  bearing, 
83. 

To  run  a  traverse,  83. 

Notes,  84. 

Angles  by,  85. 

Cautions,  85. 

Special  forms  of,  94. 

Traversing  with,  104. 

Solar,  116. 

used  with  chain,  205.  v 

in  land  surveys,  213. 
Compensating  errors,  23,  62. 
Compound  curves,  243. 
Connecting  surface  and  underground 
work,  316-320. 

Mine  entered  by  tunnel,  316. 

Mine  entered  by  two  shafts,  316. 

Mine  entered  by  one  shaft,  317. 
Constant  factors  with  slide  rule,  192. 
Construing  descriptions  of  surveys,  2 15. 
Continuous  reading  method  of  meas- 
uring angles,  255,  257. 
Contour  lines,  245,  246,  247,  248,  268. 
Contour  map,  of  conical  hill,  246. 

of  small  areas,  248. 

of  valley,  281,  282. 

of  water,  288. 
Convergence,  of  meridians,  224. 

Table  of  angular,  371. 
Cooley's  "Judicial  Functions  of  Sur- 
veyors," 341. 

Coordinates,   156-162,  170,  203,  214, 
249,  328. 

Definitions,  156. 

Elementary  problems,  157. 

To  find  the  area,  159. 

To  make  the  coordinates  all  posi- 
tive, 160. 

Elongated  areas  by  offsets,  161. 

Zero  azimuth,  162. 

Areas  determined  by,  170. 

computed  in  land  surveys,  203. 

Application  to  farm  surveys,  214. 

Polar,  249. 

Model  example  in,  328. 
Corners,  Location  and  marking  of,  209 
214,  215,  218,  228,  234,  343,  346. 
Correction  coefficient  for  temperature 
and  hygrometric  conditions,  Ta- 
ble of,  361. 


474 


INDEX. 


Correction  lines,  223. 

Corrections,   along  slopes,    Table  of, 

361. 

for  refraction,  Table  of,  366,  367. 
Cosines,  Use  of,  in  land  survey  com- 
putations, 165. 
Table  of  natural,  466,  467. 
Cotangents,  Table  of  natural,  468,  469. 
Crockett,  C.  W.,  Article  on  Slide  Rule, 

179-198. 

Cross-cut  in  mines,  309. 
Cross  profiles,  280. 

Cross-section  paper,  Use  of,  280,  281. 
Cross  sections  of  streams,  288. 
Cube  root,  Extraction  by  slide  rule, 

192. 

Culmination  of  Polaris,  90,  120,  366. 
Cumulative  or  constant  errors  in  chain- 
ing, 22,  23. 

Current  meter,  used  to  determine  ve- 
locity of  flow,  288. 
Described,  298. 
Rating  of,  300. 
Currents,  Direction  of,  304. 
Curvature  of  earth,  11,  40. 

Consideration  of,  in  leveling,  52, 

60,  62,  68. 
Curves,  238-243. 
Use  of,  238. 
Principles,  238. 
Laying  out  a  curve,  240. 
Location  by  chain  alone,  242. 
Compound  curves,  243. 
Cut,  Areas  of,  279,  280,  286. 

Datum  or  base  surface,  61,  55. 
Declination,  magnetic,  defined,  86. 

of  sun,  117. 

Change  in  magnetic,  211,  212. 

Refraction  corrections  for,  366,  367. 

See  also  "  Magnetic  declination." 
Declination  arc,  121,  123. 
Declination  vernier,  89,  122,  212,  213. 
Deeds,  Interpretation  of,  215,  235. 
Defective  sides,  150,  151. 
Departure,  old  term  for  longitude  dif- 
ference, 162. 
Depth  of  cut  or  fill.  279. 
Description  of  surveys,  Construing  of, 

215. 
Diagram  for  stadia  measurements,  134, 

135. 
Diamond  drills,  289. 


Differential  leveling,  51. 
Dip,  of  needle,  79. 

term  in  mining,  defined,  309. 
Direction,  and  measurement  of  angles, 
77-126. 

Instruments  named,  77. 

Compass,  77-79. 

Adjustments,  79-82. 

Use  of  compass,  83-85. 

Magnetic  declination,  86-94. 

The  transit,  95-99. 

Use  of  transit,  100-108. 

Adjustment  of  transit,  108-116. 

Solar  transit,  116-122. 

Adjustment  of  solar  transit,  122, 123. 

Saegmuller  solar  attachment,  123- 
125. 

Meridian  and  time  by  transit  and 

sun,  125,  126. 
Direction  meter,  288,  304. 
Direction  of  current,  how  determined, 

304. 
Discharge  of  streams  determined,  288, 

304. 

Discrepancies,   in   legal   requirements 
for  U.  S.  land  surveys,  222. 

in  surveys,  234,  236. 
Distances,  Measurement  of,  77. 
Diurnal  variations  of  declination.  87. 
Division,  of  land,  163-165. 

Occurrence  of  problem,  163. 

Solution  of  problem,  163. 

by  slide  rule,  183. 
Dorr's  "  Surveyor's  Guide,"  228. 
Double  centering,  with  transit,  102. 

in  mine  surveys,  317. 
Double    longitudes,     152,     153,     155, 

168. 

Double  sextant,  298. 
Dredging  of  river,  lake,  etc.,  275. 
Drift,  defined,  308. 

Mine  entered  by,  316. 
Dumpy  level,  43,  47,  61. 

Adjustment  of,  70. 

Earthwork  computations,  275-286. 
Ordinary  methods,  275-281. 
Estimating  volumes  from  a  map, 

281-286. 

Earthwork,  Payment  for,  275. 
Eccentricity,  115. 

Elevation,  Differences  of,  51,  134,  380- 
382. 


INDEX. 


475 


Elevation,   Determination  of,  in  stadia 
surveys,  250,  252. 

Barometric  table  of,  362,  363. 
Ellicott's  boundary  line,  357. 
Ellipse,  shape  of  longitude  meridians, 

10. 

Elongated  areas,  161. 
Elongation  of  Polaris,  90,  106,  365. 
Embankments,  Horizontal  area  of,  283. 

Reservoir  in,  285,  286. 
Engine,  Horse  power  of,  192. 
Engineer's  scale,  265. 
Engineer's  transit,  123,  255,  256. 
"Engineering  News,"  131. 
Equation  of  a  straight  line,  303. 
Equator,  9,  116,  117,  119. 
Errors,  in  measuring  level  and  hori- 
zontal lines,  22-30. 

in  chain  and  tape  measurements, 
22-30. 

Cumulative,  23. 

Compensating,  23,  62. 

in  use  of  level,  60. 

of  graduation,  115. 

in  latitude  and  altitude,  126. 

of  closure,  141,  143,  144,  146,  146, 
171. 

in  bearing,  145. 

in  latitude  and  longitude,  146,  148. 

of  geographical  position,  263. 

in  earthwork  computations,  277. 
Examples  in  land   surveys,   165-172. 
See  also  ' '  Problems. ' ' 

Logarithms,  166. 

Example  stated,  166. 

Balancing,  166. 

Latitude    differences    and    double 
longitudes,  168. 

Areas  by  coordinates,  170. 

Supplying  an  omission,  171. 
Excavations,  of  pit  or  cellar,  275. 

under  water,  281 . 

Horizontal  area  of,  283. 

Reservoir  in,  285,  286. 
Expansion  of  steel,  Coefficient  of,  24. 
Extent  of  city  surveys,  231. 
Exteriors,  Township^  225. 
External  secant,  239. 
Extinct  corners,  343-346. 
Eyepiece  of  telescope,  46. 

Focusing  of,  63. 

of  transit,  115. 

of  sextant,  296. 


Farm  surveys,  206,  206,  208-219. 


Original  surveys,  209.  [209. 

Making  original  survey  and  map, 

Description,  210. 

Resurveys,  211. 

Reasons  for  resurveys,  211. 

Procedure,  211. 

Change  of  declination,  212. 

Transit  or  compass,  213. 

Report,  214. 

Application  of  coordinates,  214. 

Principles  for  guidance  in  resur- 
veys, 216. 

Location  surveys,  219. 
Feet  changed  to  meters,  379. 
Field  methods,  for  small  area,  248. 

for  large  area,  248. 
Field  notes.     See  "Notes." 
Field  rules,  218. 

Filament  of  stream,  Vertical,  304. 
Fill,  Area  of,  279,  280,  286. 
Flag  signals,  255. 

Floats,  to  determine  direction  of  sur- 
face currents,  288. 

Rod,  303,  304. 

Flow  of  stream,  Velocity  of,  288. 
Foresights,  51,  57. 
Formulas,  for  sag  and  pull,  25,  26. 

for  measuring  on  slopes,  28. 

for  correction  for  curvature,  63. 

for  barometer  readings,  75. 

for  adjustment  of  compass,  81. 

for  azimuth  at  elongation,  92. 

for  meridian  and  time  by  transit 
and  sun,  125,  126. 

for  azimuth  of  sun,  126. 

for  stadia  measurements,  128-130. 

for  inclined  stadia  readings,  133. 

for  difference  of  elevation  by  stadia, 
134. 

for  reductions  in  stadia  work,  140. 

for  latitude  and  longitude  differ- 
ences, 142. 

for  distributing  errors  in  balancing, 
146-148. 

for  supplying  omissions,  150-162. 

for  double  longitudes,  153. 

for  finding  areas  of  closed  surveys, 
155. 

for  irregular  areas  by  offsets,  156. 

for  coordinate  computations,  160, 
161. 


476 


INDEX. 


Formulas,  for  dividing  land,  164,  165. 

for  use  of  planimeter,  175,  177-179. 

for  slide  rule1  computations,  183-196. 

for  horse  power  of  engine,  192. 

for  change  of  declination,  212,  213. 

for  angular  convergence  of  me- 
ridians, 225. 

for  curves,  239,  240,  242. 

for  prisms,  276,  279,  280. 

for  prismoids,  277,  278. 

for  estimating  volumes  from  a  map, 
283,  284. 

for  surface  grading,  279,  283. 

for  locating  soundings,  293. 

showing  theory  of  sextant,  296. 

for  correction  coefficient  for  tem- 
perature and  hygrometric  con- 
ditions, 361. 

for  latitude  of  place,  364. 

for  azimuth  of  Polaris,  92,  364. 

for  refraction  correction  to  declina- 
tion of  sun,  366. 

for  magnetic  declinations,  368-370. 

Trigonometric,  373. 

for  circular  measures,  376. 
Forward  bearing,  83,  84. 
Francis's  "  Lowell  Hydraulics,"  304. 
Fuller's  slide  rule,  197. 
Functions  of  surveyors,  341. 

Gage  for  soundings,  290. 

Gage  point,  192. 

Galilean  telescope,  45. 

Gas-pipe  rods,  289. 

Gauss  logarithmic  tables,  165. 

Geodetic  survey,  defined,  11,  12. 

Adjustment  of  triangles  in,  260. 
Geographical  positions  of  base  lines 
and    meridians  in    public  sur- 
veys, 357-360. 
German  dial,  313. 
Gila  and  Salt  River  meridian,  360. 
Grade  element  to  be  noted,  254. 
Grade  line,  denned,  282. 
Gradienter  of  tachymeter,  99. 
Grading,  of  streets,  280. 

Surface,  279,  283. 

Grading  operations,  275,  279,  280,  283. 
Great  Bear,  Constellation  of,  90. 
Guide  meridians,  224. 
Gunter's  chains,  described,  14. 

expressed  in  feet  and  meters,  377, 
378. 


Gurley's     monocular    and     binocular 
hand  levels,  73. 

Hachures,  245. 

Hand  levels,  27,  72,  73. 

Hanging  clinometer,  313. 

Harbor,  Survey  of,  294. 

Hatching,  245. 

Hill-shading,  245. 

Historical  note  of  U.  S.  Public  Land 

Surveys,  221. 
Hodgman  and  Bellows's  "  Manual  of 

Land  Surveying,"  214,  228. 
Horizontal  angles,  denned,  10. 

measured  by  transit,  108. 
Horizontal  axis   of    plane   table,  ad- 
justed, 274. 

Horizontal  distances  and  differences 
of  elevation,  Table  of,  380-382. 
Horizontal  line,  defined,  10. 
Horizontal  plane,  defined,  10. 

Generation  of,  40. 
Horse  power  of  engine,  Formula  for, 

192. 

Humboldt  meridian,  360. 
Huntsville  meridian,  358. 
Hydraulic  engineering,  287. 
Hydrographic  surveying,  287-304. 

Definition,  287 

Objects,  287. 

Work  of  surveyor,  287. 

Statement  of  methods,  288. 

Soundings,  289-2U4. 

The  sextant,  294-298. 

Measuring  velocity  discharge,  298- 
304. 

Direction  of  current,  304. 

Illumination,  of  transit  wires,  10(5. 

of  plumb  lines,  107,. 313,  314. 
Inaccessible  points,  202,  203. 
Incline  in  mine,  defined,  :!05>. 
Inclined  line,  defined,  10. 
Inclined  stadia  readings,  132. 
Index  glass  of  sextant,  295. 
Indian  meridian,  360. 
Instruments  for  surveying : 

Chains,  13. 

Tapes,  15,  17. 

Measuring  rods,  17. 

Pins,  18. 

Range  poles,  18. 

Vernier,  31. 


INDEX. 


477 


Instruments  for  surveying : 

Level  bubble,  35. 

Level,  40. 

Telescope,  43. 

Leveling  rods,  47. 

Locke  hand  level,  72,  73. 

Abney  level  and  clinometer,  72,  73. 

Gurley's  monocular  and  binocular 
hand  levels,  73. 

Barometer,  74,  76. 

Compass,  77. 

Special  forms  of  compass,  94. 

Transit,  95. 

Tachymeter,  99,  311. 

Solar  transit,  116. 

Saegmuller  solar  attachment,  123. 

Stadia,  127,  131. 

Slide  rule,  136,  179. 

Planimeter,  172. 

Mannheim  rule,  181,  198. 

Thacher  rule,  196. 

Fuller's  slide  rule,  197. 

Transit  for  city  surveying,  231. 

Protractors,  263. 

Colby's  protractor,  265. 

Ockerson's  protractor,  266. 

Plane  table,  268. 

Lead,  289. 

Sextant,  294. 

Current  meter,  208. 
.  Rod  floats,  303. 

German  dial,  313. 

Hanging  clinometer,  313. 

Lamps,  314. 

Intersections,  Method  of,  272. 
Introduction,  9-12. 

Preliminary  conceptions,  9. 

Surveying  defined,  11. 
Irregular  areas,  155,  161. 
Islands,  in  streams,  Ownership  of,  348. 

in  Great  Lakes,  349. 
Isogonic  line,  87. 

Jacob  staff,  94. 

Johnson,  J.  B.,  References  to,  25,  263. 
Johnson,  W.    D.,    Leveling  head   in- 
vented by,  269. 
Judicial  functions  of  surveyors,  341- 

350. 

Imperfect  training,  341. 
Duties,  regarding  location  of  mon- 
uments, 342,  344. 
Relocation  of  extinct  corntrs,  :}(:}. 


Judicial  functions  of  surveyors : 

Corners  not  to  be  established,  343. 
Must  regard  occupation  and  claims, 

344. 

Must  ascertain  facts,  346. 
Surplus  and  deficiency  in  appor- 
tionment, 346. 

Difficulties  with  meander  lines,  34(5. 
Extension  of  water  fronts,  348. 
Bed  ownership  on  water  fronts,  348. 
Riparian  rights  on  lakes,  349. 

Key  to  topographical  symbols  on  niups, 
267. 

Lakes,  Riparian  rights  in,  217,  349. 

Meandering  of,  228. 

Dredging  of,  275. 
Land,  Map  of,  11. 

Division  of,  163. 

Title  to,  343. 
Land  survey  computations,  141-198. 

Considerations  and  definitions,  141. 

Error  of  closure,  143. 

Balancing  the  survey,  144-148. 

Supplying  omissions,  149-152. 

Areas,  152-156. 

Coordinates,  156-162. 

Dividing  land,  163-165. 

Model  examples,  165-172. 

The  planimeter,  172-179. 

The  slide  rule,  179-198. 
Land  surveys,  201-237. 

Obstacles  and  problems,  201-203. 

Two  common  problems,  204. 

Surveying  with  chain  alone,  204- 
208. 

Farm  surveys,  208-219. 

United  States  public  land  surveys, 
219-229. 

City  surveying,  230-237. 
Lateral  adjustment,  69. 
Latitude,  Parallels  of,  10. 

Observation  for,  121. 

of  a  line  and  point,  defined,  142. 

differences,  142,  147-152,  155,  162, 
168,  204. 

Errors  in,  146,  148. 

and  longitude  plotting,   209,  210, 
261,  301,  306. 

Table  of  lengths  of,  372. 

Formula  for,  364. 
Laying  out  curves,  240. 


478 


INDEX. 


Lead  used  for  soundings,  288,  289. 
Least  squares,  formula  from  Treatise 
on,  30. 

Reference  to  demonstration  of  rule 
for  balancing  survey  by,  147. 

Method  of,  mentioned,  303. 
Legal  requirements   for    public    land 

surveys,  222. 
Length,  Standard  of,  231. 
Lesser  Bear,  Constellation  of,  90. 
Lettering  on  maps,  210,  267. 
Level  and  horizontal  lines,  Measure- 
ment of,  13-30. 

Level  line,  defined,  10. 

Line  to  be  measured,  13. 

Instruments  used,  13-18. 

Methods,  18-22. 

Classes  of  errors,  22. 

Causes  of  errors,  23. 

Temperature,  24. 

Sag  and  pull,  25,  26. 

Alignment,  27. 

Slope,  27. 

Precision  to  be  obtained,  29. 
Level  bubble,  35-38. 
Level,  term  in  mining,  defined,  308. 
Level  tube  for  chaining,  27. 
Level  under  telescope,  114. 
Level,  Use  and  adjustment  of,  50-71. 

Adjustment  and  setting  up,  50. 

Differential  leveling,  51. 

Profile  leveling,  53. 

Making  the  profile,  58. 

Leveling  over  an  area,  60. 

Errors,  60. 

Curvature  and  refraction,  62. 

Reciprocal  leveling,  63. 

Focusing,  63. 

Adjustments  named,  64. 

Collimation  adjustment,  64. 

Adjustment  of  objective  slide,  65. 

Bubble  adjustment,  67. 

Peg  method,  68. 

Lateral  adjustment,  69. 

Y  adjustment,  70. 

Adjustment  of  dumpy  level,  70. 

Adjustment  of  any  level  on  a  metal 
base,  71. 

Hand,  72,  73. 
Level  vial,  36,  37. 

Leveling  or  measuring  differences  of 
altitude,  40-76,  308. 

General  principle,  40. 


Leveling  or  measuring  differences  of 
altitude,  Instruments  for,  40-50. 

Use  of  level,  50-63. 

Adjustments  of  level,  63-71. 

Minor  instruments,  72,  73. 

Leveling  with  barometers,  74-76. 

in  underground  surveys,  308. 
Leveling  rods,  47-50. 
Limb  of  transit,  96. 
Line  of  collimation,  defined,  46. 

in  transit,  108. 

Revolution  of,  109. 

Linear  transformations,  Table  of,  377. 
Linen  tapes,  17. 
Local  attraction  of  magnetic  needle, 

93. 
Local  mean  times  of  culmination  and 

elongation  of  Polaris,  365. 
Locating  arcs  by  chain,  242. 
Location  surveys,  208,  219. 
Locke  hand  level,  72,  73. 
Lode  claim,  306. 

Lodes,  Coloring  of,  on  maps,  320. 
Logarithms,  Computations  by,  in  land 
surveys,  165. 

Scale  of,  defined,  180. 

Use  of  scale,  181. 

Infinite  extent  of  scale,  184. 

Scale  of  logarithmic  sines  and  tan- 
gents, 193,  194. 

Table  of  numbers,  397-418. 

Table  of  trigonometric  functions, 

419-464. 
Longitude,  Meridians  of,  10. 

of  a  line  and  point,  defined,  142. 

Differences  of,  142,  147-1. r>0,  151, 
153,  162,  204. 

Errors  in,  148. 

Double,  152,  153,  155,  1«8. 

Table  of  lengths  of,  372. 
Lot,  Description  of,  233. 
Louisiana  meridian,  359. 
Lunar  variation  of  declination,  86. 

Magnetic  bearings,  212. 

Magnetic  declination,  86-94,  368-:57<). 

defined,  86. 

Variations  of,  86. 

Determination  of  declination,  87. 

Determination  of  true  meridian,  89. 

Azimuth  at  elongation,  91. 

Local  attraction,  93. 

Special  forms  of  compasses,  94. 


INDEX. 


479 


Magnetic  declination,  Table  of,  368- 

370. 

Magnetic  meridian,  78, 86, 103, 107,  211. 
Magnetic  needle,  77,  78,  364. 
Mannheim  rule,  181,  198. 
"Manual  of  Surveying  Instructions," 

221,  228,  365. 
Map  lettering,  210,  267. 
Mapping,  261-268,  320,  321. 

Triangles,  261. 

Outline  of  method  for  topography, 
261. 

Stadia  line,  262. 

Side  shots,  263. 

Colby's  protractor,  265. 

Ockerson's  protractor,  266. 

Finishing  the  map,  267. 

Requirements  for  maps,  267. 

Scale,  268. 

for  metal  and  coal  mines,  320. 

Scale  in  mines,  321. 

Problems,  321. 
Maps,  defined,  11. 

of  farm  surveys,  209. 

of  city  surveys,  237. 

Topographical,  244. 

Requirements  for,  267,  354-356. 

Contour  for  grading,  281,  282. 

of  harbor  surveys,  294. 

of  mines,  320. 

for  public  records,  353. 
Maximum  velocity  of  streams,  298. 
Mean  surface  of  earth,  9. 
Meander  corners,  228. 
Meander  lines,  346,  347,  349. 
Meandering  a  stream,  227. 
Measuring  differences  of  altitude.    See 

"Leveling." 
Measuring  velocity  discharge,  298-304. 

Position  of  maximum  velocity,  298. 

Current  meters,  298. 

Use  of  meter,  300. 

Rating  the  meter,  300. 

Rod  floats,  303. 

Discharge,  304. 
Mercurial   barometer,  Leveling  with, 

74,  76. 
Meridians,  of  longitude,  10. 

Magnetic  and  true,  86,  89,  225. 

Determination  of,  105. 

Principal,  in  U.  S.  land  surveys,  219, 
223,  357-360. 

Angular  convergence  of,  224,  371. 


Meridians,  Guide,  224. 

of  reference,  261. 

Table  of  distance  between,  371. 
Meridian  and  time,  by  transit  and  sun, 

125,  126. 

Metal  base,  Adjustment  of  level  on,  71. 
Metal  mines,  306,  320. 
Metallic  tapes,  17. 
"Metes  and  bounds,"  Description  by, 

216,  235. 

Methods  for  earthwork  computations, 
275-281. 

Occurrence  of  problem,  275. 

Prisms,  275. 

Prismoids,  276. 

Prismoidal  formula,  277. 

Approximations,  277. 

Area  grading,  279. 

Street  grading,  280. 

Excavation  under  water,  281. 
Methods  of  measuring  level  and  hori- 
zontal lines,  18-22. 

Preliminary  statement,  18. 

Chaining,  18. 

Hints,  21. 

Chaining  on  slopes,  21. 
Meters  changed  to  feet,  379. 
Michigan  meridian,  358. 
Michigan,  Survey  of  lands  in,  341. 
Micrometer  screw,  36. 
Mine  surveying,  305-321. 

Surface  surveys,  305-308. 

Underground  surveys,  308-316. 

Connecting    surface     and    under- 
ground work,  316-320. 

Mapping  the  survey,  320,  321. 
Mining  claim,  Form  of,  306. 
Minus  sights,  51,  56,  57,  60. 
Model  examples.    See  "Problems." 
Modulus  of  elasticity  of  chains,  25. 
Monocular  hand  level,  73. 
Montana  meridian,  360. 
Monuments  for  surveys,  209,  211,  216, 
218,  234,  237,  253,  307,  341,  344, 
345,  351,  355. 

Mount  Diablo  meridian,  369. 
Multiplication  by  slide  rule,  182. 
Myers's,  John  H.,  problems  in  coordi- 
nates, 328. 

Natural  scale  of  maps,  268. 
Natural  sines  and  cosines,  Table  of, 
466,  407. 


480 


INDEX. 


Natural  tangents  and  cotangents,  Table 

of,  468,  469. 

"Nautical  Almanac,"  120,  126. 
Needle  and  pivot  of  compass,  80. 
Needle  checks  on  azimuths,  107. 
New  Almaden  mine,  310. 
New  Mexico  meridian,  359. 
New  York  leveling  rod,  47,  48. 
North  Star,  Observation  on,  90. 
Notes,  in  profile  leveling,  55. 

in  traversing,  84. 

in  stadia  measurements,  140. 

for  land  surveys,  169,  208,  228,  229. 

for  topography,  250,  252. 

in  triangulation,  256. 

for  mine  surveys,  307,  320. 

for  underground  work,  315. 

Private,  351,  364. 

Objective  of  telescope,  43,  45,  46. 
Objective  slide,  Adjustment  of,  65. 
Oblate  spheroid  of  revolution,  9. 
<  )bstacles  in  land  surveys,  201. 
Ockerson's  protractor,  266. 
Offsets,  155,  156,  161. 
Omissions,   supplied    in    land  survey 
computations,  149-152,  171. 

Necessity  for,  149. 

General  discussion,  149. 

Cases  I. -VI.,  150. 

Algebraic  solution,  151. 
Ordinates,  156,  160,  161. 
Ore,  Mining  of,  308. 
Orientation,  of  transit,  104,  261,  273. 

of  plane  table,  273. 
Origin  of  coordinates,  156. 
Original  land  surveys,  208,  209-211. 
Ormsbee,  J.  J.,  on  three-tripod  system, 

315. 
Oughtred,  inventor  of  sliding  scales, 

180. 
Ownership  of  surveys,  351-356. 

Parallelepiped,  Truncated,  276. 
Parallels,  of  latitude,  10. 

Standard,  223. 
Park  drive  curves,  238. 
Partridge's  improvement  of  slide  rule, 

180. 

Patenting  mining  claims,  307. 
Peg,  a  turning  point,  57. 
Peg  method  of  adjusting  telescopes,  68, 

71,  274. 


Philadelphia  leveling  rod,  47,  48,  49. 

Pins  used  in  chaining,  18. 

Pitch,  defined,  309. 

Pivot  of  compass,  80. 

Placer  claim,  306. 

Plane  survey,  denned,  12. 

Plane  table,  268-274,  294. 

Description,  268. 

Use,  270. 

Three-point  and  two- point    prob- 
lems, 272. 

Adjustments,  274. 

used  in  harbor  surveys,  294. 
Planimeter,  172-179,  270,  284. 

Description,  172. 

Use,  173. 

Theory,  174. 

Zero  circumference,  178. 

Circumference  of  wheel,  178. 

Length  of  arm,  179. 

used  in  measuring  areas,  270,  284. 
Plate  bubbles,  of  compass,  79. 

of  transit,  108. 
Plotting,  of  maps,  209. 

by  latitudes  and  longitudes,  209, 
210,  261,  301,  306. 

of  corners,  210. 

of  points  on  maps,  292. 
Plumb  line,  a  vertical  line,  10. 

Use  of,  22,  90. 
Plumbing,  253. 
Plummet  lamps,  314. 
Plummets  for  mine  surveys,  318,  319. 
Plus  sights,  61,  60. 
Polar  axis  of  solar  transit,  123. 
Polar  coordinates,  249. 
Polar  distance  of  Polaris,  364. 
Polar  planimeter,  172,  173. 
Polaris,  Observations  on,  90. 

Elongation  and  culmination  of,  90, 
106,  107,  120,  223,  365. 

Declination  of,  92. 

Polar  distance  of,  92,  364. 
Porro,  inventor  of  stadia  rod,  127. 
Positive  coordinates,  160. 
Precise  level,  43,  47,  61. 
Precision  necessary,  in  measurement, 
29. 

in  city  surveying,  230. 
Price,  W.  G.,  Observations  by,  301,  339. 
Principal  meridians,  219,  223,  357-360. 
Prisms,  60,  275-277. 
Prismatic  compass,  94. 


INDEX. 


481 


Prismoids,  276. 

Prismoidal  formula,  276,  277,  279, 283, 

284. 

Private  notes  of  survey,  351,  354. 
Problems  and  examples,  157-159, 165- 
172,  186,  201-204,  321,  322-340. 

in  coordinates,  157-159,  328. 

in  land  surveys,  165-172,  201-204. 

with  slide  rule,  186. 

in  mine  surveying,  321. 

on  Chapter  1.^  322. 

on  Chapters  III.  and  IV.,  324. 

on  Chapters  V.  and  VI.,  326. 

on  Chapter  VIII.,  335. 

on  Chapter  IX.,  336. 

on  Chapter  X.,  338. 

on  Chapter  XI.,  339. 

on  Chapter  XII.,  340. 
Profile  leveling,  53-60. 
Profile  paper,  58,  59. 
Profiles,  Cross,  280. 
Proportion  by  slide  rule,  183. 
Protractor,  Land  survey  drawing  made 
by,  141. 

Vernier,  263. 

Colby's,  265. 

Ockerson's,  266. 

Three-arm,  292,  293. 

Plotting  with,  306. 
Public  surveys,  in  Michigan,  341. 

Base  lines  governing,  357-360. 

See  also  "U.  S.  public  land  sur- 
veys." 
Pull,  Formula  for,  25. 

Elimination  of,  26. 

element  noted,  254. 
Pull  scale,  231. 
Pyramid,  276,  277. 

Quarter  sections,  220. 

Rack  and  pinion  movement  of  tele- 
scope, 44. 

Radiation,  Method  of,  270. 

Railroads,  Surveys  of,  238,  268. 
Curves  on,  238. 

Random  lines,  203,  225,  227. 

Random  surveys,  214,  215. 

Range  lines,  220,  225,  226,  227,  291. 

Range  poles,  18. 

Raymond's  (R.  W.)  "  Glossary  of  Min- 
ing and  Metallurgic  Terms,"  308. 

Reading  glasses,  98. 

H'M'U  SURV.  — ;J1 


Reciprocal  leveling,  63. 

Reed's  "Topographical  Drawing  and 

Sketching,"  267. 
Reels,  15-17. 
Reference  lines,  223. 
Reflector  in  telescope,  106. 
Refraction,  Effect  of,  63,  139. 
Corrections  for,  121,  125,  126. 
Table  of  corrections  for,  364,  366, 

367. 
Repetition    measurement   of    angles, 

255,  256. 

Report  of  survey,  214. 
Report  of  U.  S.  Coast  and  Geodetic 

Survey,  75,  86,  87,  368. 
Reservoir,  Building  of,  275. 
Capacity  of,  281. 
in  embankment   and   excavation, 

285,  286. 

Resurveys,  208,  211-218. 
Riparian  rights,  217,  347,  349. 
Ritchie  and  Haskell's  direction  meter, 

304. 

River,  Dredging  of,  275. 
River,  Survey  of,  294. 
Roads,  Survey  of,  238. 
Rod  floats,  Velocities  of,  303. 
Rod  level,  61. 

Rods,  Wooden  and  metallic,  13,  17. 
Gas-pipe,  289. 
for  stadia  measurements,  131,  136- 

138. 

Rolling  planimeter,  172,  173. 
Runner,  in  slide  rule,  180,  185,  191. 

Saegmuller  solar  attachment,  123. 
Sag,  Formula  for,  25. 

Elimination  of,  26. 
Saint  Helena  meridian,  359. 
Saint  Stephen's  meridian,  358. 
Salt  Lake  meridian,  359. 
Salt  River  meridian,  360. 
San  Bernardino  meridian,  360. 
Scale,  of  slide  rule,  179. 

Logarithmic,  180-195. . 

of  logarithmic  sines  and  tangents, 
193, 194. 

of  topographical  maps,  268. 

of  mine  maps,  321. 
Schott's  formulas  for  declination,  87. 
Scow,  Displacement  of,  281. 
Sea  level,  246. 
Secant,  External,  239. 


482 


INDEX. 


Sections,  220. 

Secular  variations  of  declination,  86. 

Sequence  of  figures,  in  operations  with 

slide  rule,  185. 
Seven  Ranges,  221. 
Severn  tunnel,  320. 
Sextant,  77,  294-298. 

use  mentioned,  77. 

Description  of,  294. 

Use  of,  295. 

Theory  of,  296. 

Adjustments  of,  297. 

Other  forms  of,  298. 
Shafts,  defined,  308. 

Mine  entered  by,  317,  318. 
Shifting  center  of  transit,  99. 
Side  shots,  249,  263. 
Sights  of  compass,  80. 
Similar  figures,  Areas  of,  277. 
Similar  triangles,  128. 
Simple  triangulation,  253-261. 

when  used,  253. 

Measuring  base  line,  253. 

Measuring  angles,  254. 

Notes,  256. 

Adjusting  triangles,  259. 

Computing  triangles,  260. 

Use  of  triangles,  260. 
Sines  and  cosines,  165,  193. 

Table  of  logarithmic,  419-464. 

Table  of  natural,  466,  467. 
Sketch  of  tract  to  be  surveyed,  205. 

for  topography,  251. 
Slide  direct,  186. 

inverted,  189. 

reversed,  193. 

Slide  rule,  134,  136,  140,  179-198,  253, 
281,  380. 

used  in  stadia  readings,  134,  136, 
140,  380. 

for  differences  of  elevation,  136. 

described,  179. 

Historical  note  of,  180. 

Construction  of  scales,  180. 

Use  of  scales,  181. 

Mannheim  rule,  181. 

Use  of  rule,  182. 

Extent  of  logarithmic  scale,  184. 

Sequence  of  figures,  185. 

Shifting  the  slide,  185. 

Use  of  runner,  185. 

Squares  and  square  roots,  185. 

Statement  of  problems,  186, 


Slide  rule,  Slide  direct,  186. 

Slide  inverted,  189. 

Use  of  runner  in  complicated  ex- 
pressions, 191. 

Gage  points,  192. 

Extraction  of  cube  roots,  192. 

Slide  reversed,  193. 

Scale  of  logarithmic  sines,  193. 

Scale  of  logarithmic  tangents,  194. 

The  Thacher  rule,  196. 

Settings  for  Thacher  rule,  197. 

Fuller's  slide  rule,  197. 

Mannheim  rule,  198. 

used  in  volumetric  computations, 

281. 
Slopes,  Chaining  on,  27. 

Formula  for  measuring  on,  28. 

in  mines,  309. 

Smith's  observations  for  stadia  meas- 
urements, 139. 

Smithsonian  geographical  tables,  372. 
Solar  attachment  of  transit,  122. 
Solar  compass,  77, 116. 
Solar  transit,  1 16-125.    See  "  Transit." 
Soundings,  288-294. 

in  hydrographic  surveying,  288. 

Making  soundings,  289. 

Locating  soundings,  290. 

Occurrence  of  methods,  293. 

Survey  of  harbor,  river,  etc.,  294. 
Sounding  lead,  288. 
Spacing  of  stadia  wires,  129. 
Spanish  vara,  221. 
Spider  lines,  44. 
Spirit  level,  35.        jV;. 
Squares  and  square  roots  by  slide  rule, 

185. 

Stadia  line,  Plotting  of,  261,  262,  263. 
Stadia  measurements,  127-140. 

defined,  127. 

Method  explained,  128. 

Spacing  of  the  wires,  129. 

To  approximate  the  value  of  /,  130. 

The  value  of  c,  130. 

The  value  of  |and  (/  +  c),  130. 

The  rod,  131. 

Inclined  readings,  132. 

Table  of,  133. 

Difference  of  elevation,  134. 

Diagram,  134. 

Slide  rule,  136. 

Graduating  a  stadia,  136, 


INDEX. 


483 


Stadia  measurements,  Smith's  obser- 
vations, 139. 

Notes,  140. 

Stadia  method  of  topography,  249. 
Stadia  reduction  table,  380-382. 
Stadia  surveys  in  coal  mines,  306. 
Stadia  traverse,  261. 
Stadia  wires  of  tachy meter,  99. 
Standard  length,  231. 
Standard  parallels,  223. 
Stanley's   "  Surveying    and    Leveling 

Instruments,"  71. 
Station  marks  in  mines,  309,  310. 
Statutes  for  surveyors,  Need  of,  356. 
Steel,  Coefficient  of  expansion  of,  24. 
Steel  tapes,  15,  24,  313. 
Slope,  denned,  309. 
Streams,  Meandering  of,  227. 

Cross  sections  of,  288. 
Street  grading,  280. 
Striding  level,  71,  109. 
Strike,  defined,  309. 
Structures,  Volume  of,  285. 
Stiibben,  J.,  Paper  by,  237. 
Subcurrents,  Direction  of,  288,  304. 
Subdivision  in  land  surveys,  219. 

of  townships,  226. 
Sun,  Declination  of,  117,  125,  126. 
Surface   currents,    Direction  of,   288, 

304. 

Surface  form,  Representation  of,  244. 
Surface  grading,  281,  283. 
Surface  monuments  of  mines,  307. 
Surface  of  earth,  9,  11. 
Surface  surveys,  305-308. 

Coal  mines,  305. 

Metal  mines,  306. 

Form  of  mining  claim,  306. 

Surveying  the  claim,  307. 

Surface  monuments,  307. 
Surveying,  defined,  11,  351. 
Surveyors,     Difference     of      opinion 
among,  351. 

Laxity  and  dishonesty  of,  352. 

Imperfect  maps  of,  353,  354. 

Duties  regarding  maps,  355,  356. 
Surveys,  Public,  in  Michigan,  341.    See 

"U.  S.  public  land  surveys." 
Suspended  planimeter,  172,  173. 

Tables,  361-469 : 

I.  Correction  to  100  units  measured 
along  slopes  given,  361. 


II.  Correction  coefficient  for 
temperature  and  hygro- 
metric  conditions,  3(51. 

III.  Barometric  elevations,  362. 

IV.  Polar  distance  of  Polaris, 

364. 
V.  Daily  variation  of  magnetic 

needle,  364. 

VI.  Approximate  local  mean 
times  of  elongation  and 
culmination  of  Polaris, 
365. 

VII.  Refraction  corrections,  366. 
VIII.  Magnetic  declination  formu- 
las, 368. 

IX.  Angular  convergences  and 
distances  between  merid- 
ians, 371. 

X.  Length  of  one  minute  of  lati- 
tude and  longitude,  372. 
XI.  Trigonometric  functions  and 

formulas,  373. 
XII.  Lengths  of  circular  arcs,  376. 

XIII.  Linear  transformations,  377. 

XIV.  Horizontal  distances  and  dif- 

ferences of  elevation,  380. 
XV.  Common  logarithms  of  num- 
bers, 397-418. 

XVI.    Logarithms  of  trigonomet- 
ric functions,  419-4(54. 
XVII.  Natural  sines  and  cosines, 

466,  467. 

XVIII.  Natural   tangents   and    co- 
tangents, 468,  469. 
Tacheometer,  127. 
Tachymeter,  described,  99. 

term  for  complete  transit  equipped 

for  stadia  work,  127. 
used  in  mine  surveys,  311,  312. 
Tallahassee  meridian,  358. 
Tangents,  Logarithmic  scale  of,  194. 
Intersection  of,  239. 
Table  of  natural,  468,  469. 
Tapes,  used  in  measuring,  13. 
Steel,  15. 
Ribbon,  16. 
Linen,  17. 
Errors  in,  22,  23. 
Expansion  of,  24. 
used  in  land  surveys,  204. 
used  in  city  surveys,  231. 
used  in  measuring  base  line,  254. 
used  in  mine  surveys,  313,  317. 


484 


INDEX. 


Target  lamp,  314. 

Targets  of  leveling  rods,  47-49. 

Telescope,  of  level,  43. 

Spider  lines  of,  44. 

Field  of  view  of,  45. 

Eyepiece  of,  46. 

Spherical  aberration  of,  46. 

Axis  of  revolution  of,  114. 

Magnifying  power  of,  230. 

pointing  in  triangulation,  256. 
Temperature  correction  of  tapes  and 

chains,  24. 

Temperature  element  noted,  254. 
Temperature  correction,  Table  of,  361. 
Texas,  Public  lands  of,  221. 
Thacher  rule,  182,  196,  197. 
Theodolite,  123. 
Theory  of  planimeter,  174-178. 
Three-arm  protractor,  292,  293. 
Three-point  problem,  272,  273,  292. 
Three-tripod  system,  316. 
Time,  by  transit  and  sun,  125,  126. 
Topographical  surveying,  11, 127,  244- 
274,  305. 

denned,  11. 

Use  of  stadia  in,  127. 

Topography,  244-253. 

Simple  triangulation,  253-261. 

Mapping,  261-268. 

The  plane  table,  268-274. 

in  mine  surveying,  305. 
Topography,  244-253,  288. 

Methods  of   representing   surface 
form,  244. 

Field  methods  for  small  area,  247. 

Contour  map,  248. 

Field  methods  for  large  area,  248. 

Transit  and  stadia  method,  249. 

of  bed  of  water,  288. 
Township,  a  subdivision  of  public  land, 
220. 

Subdivisions  of,  226. 

Theoretical,  226. 
Township  exteriors,  225. 
"Transactions  American  Society  Civil 

Engineers,"  237. 
Transformations,  Linear,  377-379. 

Gunter's  chains  to  feet,  377. 

Gunter's  chains  to  meters,  378. 

Feet  to  meters,  379. 
Transit,  77,95-126,  148,  205,  213,  231, 
249,   255,   256,   261,    311,   312, 
317-319. 


Transit,  Description,  77,  95. 

Tachymeter,  99. 

Carrying  transit,  100. 

Setting  up,  101. 

To  produce  a  straight  line  with, 
101. 

Measuring  angles  with,  102. 

Azimuth,  102. 

Traversing,  103. 

used  instead  of  compass,  104. 

Determining  meridian,  105. 

Needle  checks  on  azimuths,  107. 

Requirements  for  adjusted,  108. 

Plate  bubbles,  108. 

Line  of  collimation,  109. 

Level  under  telescope,  114. 

Vertical  circle,  115. 

Eyepiece,  115. 

Eccentricity,  116. 

Solar  transit  explained,  116. 

Fundamental  conception  for  solar, 
116. 

Description  of  solar,  118. 

Method  of  use  of  solar,  119. 

Limitations  of  solar,  120. 

Latitude,  121. 

Refraction,  121. 

Adjustments  of  solar,  named,  122. 

Lines  of  collimation  of  solar,  122. 

Declination  vernier  of  solar,  122. 

Polar  axis  of  solar,  123. 

Engineer's,  123,  255,  256. 

Saegmuller's  solar  attachment,  de- 
scribed, 123. 

Adjustments  of  Saegmuller's,  125. 

used  in  land  surveys,  148,  213. 

used  with  chain,  205. 

City  surveyor's,  231. 

Orientation  of,  261. 

used  in  mine  surveys,  311, 312, 317- 

319. 

Transit  and  stadia  method  of  topogra- 
phy, 249. 

Traverse,  Running  of,  83. 
Traverse  tables,  165. 
Traversing,  with  compass,  83. 

with  transit,  103. 

with  plane  table,  270. 

in  underground  surveys,  308. 
Triangles,  Similar,  128. 

Division  of  fields  into,  206. 

Adjustment  of,  259. 

Computing,  260. 


INDEX. 


485 


Triangles,  used  in  mapping,  261. 

Solution  of,  373-376. 
Triangulation,  Simple,  253. 

Stations,  254,  260,  272,  294. 

System  in  topography,  272. 

System  in  coal  mine  surveys,  305. 
Trigonometer,  306. 
Trigonometric  functions,  Signs  of,  162. 

Formulas  of,  373-376. 

Tables  of,  419-469. 
Tropic  of  Cancer,  Extension  of,  117. 
Tropic  of  Capricorn,  Extension  of,  117. 
Tunnel,  defined,  308. 

Mine  entered  by,  316. 
Turning  points  in  leveling,  63. 
Two-point  problem,  272,  273. 

Underground  surveys,  308-316. 

General  statement,  308. 

Definitions,  308. 

Location  and  form  of  station  marks, 
309. 

Instruments  used,  310. 

Devices  for  making  stations  visible, 
313. 

Notes,  315. 

United  States  Coast  and  Geodetic  Sur- 
vey, References  to,  75,  86,  231, 
268,  270,  361,  362,  365,  368. 
United  States  deputy  surveyors,  219. 
United  States  Geological  Survey,  Ref- 
erences to,  268,  270. 
United    States   public   land    surveys, 
219-229. 

Character  of  work,  219. 

Scheme  of  subdivision,  219. 

Historical  note,  221. 

Legal   requirements,    inconsistent, 
222. 

Principal  reference  lines,  223. 

Standard  parallels,  223. 

Guide  meridians,  224. 

Angular  convergence  of  two  merid- 
ians, 224. 

Township  exteriors,  225. 

Subdivisions  of  townships,  226. 

Meandering  a  stream,  227. 

Corners,  228. 

Notes,  228. 
United  States  surveyor  general,  221. 


Vara,  Spanish  measure,  221. 
Variations  of  declination,  secular,  an- 
nual, lunar,  and  diurnal,  86. 
Vega  logarithmic  tables,  165. 
Veins  in  mines,  306. 
Velocity  of  flow  of  stream,  288. 
Vernier  and  level  bubble,  31-39. 
Vernier,  defined,  31. 

Direct,  31,  32. 

Retrograde,  33. 

Double,  34. 

Declination,  122. 
Vernier  protractor,  263. 
Vertical  angle,  line,  and  plane,  defined, 

10. 

Vertical  circle  of  transit,  115. 
Volumes  estimated  from  a  map,281-286. 

A  reservoir,  281. 

Application  to  surface  grading,  283. 

Application  to  structures,  286. 
Volume  of  earth,  measured,  275. 

in  street  grading,  280. 

in  mines,  308. 

See    also    "Earthwork    computa- 
tions." 

Wagon  roads,  Survey  of,  238. 

Washington  meridian,  359. 

Water  boundaries,  Statutes  regarding, 
216,  217. 

Water,  Excavation  under,  281. 

Wedges  in  earthwork,  276,  277. 

Weighting  courses,  148. 

Weir  measurements,  288. 

Willamette  meridian,  360. 

Winslow's  stadia  reduction  table,  380. 

Winze,  defined,  309. 

Wires,  Illumination  of,  106,  311. 

Witness  points,  209,  228. 

Wright's  "Adjustment  of  Observa- 
tions," 147. 

Y  adjustment,  64,  70. 
Y  level,  43,  47,  61. 

Zenith-pole  arc,  92. 
Zenith-star  arc,  92. 
Zero  azimuth,  162. 
Zero  circumference,  178. 
Zeta  Ursse  Majoris,  90. 


Typography  by  J.  S.  Gushing  &  Co.,  Norwood,  Mm. 


. 


UNIVERSITY  OF  CALIFORNIA  AT  LOS  ANGELES 
THE  UNIVERSITY  LIBRARY 


This  book  is  DUE  on  the  last  date  stamped  below 

WAR  7     19601 

"91979 


17  J938 


,JUfl? 

MAR  3     1942 
JUN  14  Wp 
SEP  101948 


JUN  19 1952! 

JUN23RECD 

NOV    31952 


Form  L-9-15m-3,'34 


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C'DYRL  FEB,2,2t)0' 


UNIVERSITY  of  CALIFORNIA 

AT 

LOS  ANGELES 
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UCLA-Young   Research    Library 

TA545   .R21t 


L  009  586  018  5 


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AA    001  270148    8 


